UC-NRLF 


LOGARITHMIC 


AND    OTHER 


MATHEMATICAL    TABLES 


BY  WILLIAM  J.  HUSSEY 

SOMETIME  ASSISTANT  IN  THE  UNITED  STATES  NAUTICAL  ALMANAC  OFFICE;  ACTING  DIRECTOR  OF 

THE  OBSERVATORY  AT  ANN  ARBOR  AND  INSTRUCTOR  IN  ASTRONOMY  IN  THE  UNIVERSITY 

OF  MICHIGAN;  ASSISTANT  PROFESSOR  OF  ASTRONOMY  (ELECT)  IN  THE  LELAND 

STANFORD  JUNIOR  UNIVERSITY,  PALO  ALTO,  CALIFORNIA. 


SECOND   EDITION. 


ANN  ABBOK,  MICHIGAN: 

THE  REGISTER  PUBLISHING  COMPANY, 

TLbe  flnlant)  Ipvess. 

1892. 


^ih-^Ti^* 


teTRONOMY  UBRARg 

Copyright,  1891,  "by 

THE  REGISTER  PUBLISHING  COMPANY. 
Ann  Arbor,  Michigan. 

First  Printed,  Oct.  1891. 
Second  Edition  with  corrections  and  additions,  March 


'• 


PREFACE.  yi 

The  extended  calculations  required  by  some  of  the  applications  of  trigonom- 
etry, are  laborious  even  to  experienced  computers;  to  beginners  they  are  often  a 
fruitful  source  of  discouragement.  Experience  in  making  calculations  and  famil- 
iarity with  the  formulas  employed  will  suggest  those  methods  of  arrangement  by 
which  skillful  computers  shorten  their  work  and  save  much  of  their  time.  It  should 
always  be  the  aim,  to  secure  the  results  to  the  degree  of  accuracy  required,  by  a 
minimum  expenditure  of  time  and  labor.  So  far  as  the  mechanical  part  of  the  work 
is  concerned,  the  principal  factors  leading  to  this  end,  are,  the  proper  arrangement 
of  the  formulas  to  be  used  before  the  computation  is  begun,  the  use  of  conveniently 
arranged  tables,  containing  needed  helps  for  facilitating  interpolation,  and  the  use 
of  no  more  places  of  decimals  than  are  necessary  to  secure  the  desired  accuracy  in 
the  results. 

Orderly  arrangement  is  almost  indispensable  to  correct  and  rapid  computation, 
and,  consequently,  the  practice  of  making  computations  on  loose  scraps  of  paper, 
without  systematic  arrangement,  should  not  be  followed.  In  the  beginning  an 
outline  of  the  entire  solution  should  be  made  by  writing  the  symbols  of  the  quanti- 
ties to  be  used  in  a  vertical  column.  Those  which  are  to  be  combined,  as  shown  by 
the  formulas,  should  be  placed  adjacent.  In  the  same  solution,  turning  more  than 
once  to  the  same  place  in  the  tables  should  be  avoided.  This  can  be  done  by  taking 
from  the  tables  at  one  opening,  all  the  functions  of  a  given  angle,  which  are  required 
in  the  solution,  and,  writing  them  in  their  proper  places. 

The  logarithmic  and  other  tables  employed  should  be  conveniently  arranged. 
They  should  contain  the  auxiliary  tables  of  proportional  parts  on  the  margins  of  the 
pages,  excepting  where  the  differences  are  so  small  that  the  interpolations  can  easily 
be  made  mentally  without  them. 

The  number  of  places  of  decimals  to  be  used  in  any  computation,  will  depend 
upon  character  of  the  data  employed  and  also  upon  the  degree  of  accuracy  required 
in  the  results.  Where  the  data  are  given  with  great  precision  and  the  results  to  be 
derived  from  them,  are  required  with  extreme  accuracy,  tables  to  seven  and  in  rare 
cases  even  to  ten  or  more  places  of  decimals  must  be  used.  But  for  nearly  all  ordi- 
nary calculations  such  precision  is  not  required,  and  the  accuracy  of  the  results 
obtained  by  the  use  of  logarithms  to  five  places  of  decimals,  is  amply  sufficient.  The 
use  of  this  number  affords  results  which  are  usually  correct  to  one  ten-thousandth 
part.  In  calculations  where  this  degree  of  accuracy  is  not  required,  a  smaller  num. 
ber  of  places  of  decimals  should  be  used.  In  such  cases  it  is  frequently  more 
convenient  to  use  natural  numbers  and  the  natural  trigonometric  functions  instead 
of  their  logarithms. 

M57W78 


V  PREFACE . 

In  compiling  this  book  of  tables  for  general  use,  the  needs  of  students  and  of 
computers  have  been  kept  in  view.  In  selecting  the  arrangements  of  the  tables, 
those  have  been  taken  which  experienced  computers  lind  most  convenient.  They  are, 
at  the  same  time,  those  which  are  best  adapted  to  the  needs  of  students.  The  book 
contains  a  large  number  of  useful  tables,  and,  it  is  believed,  that  all  needed  helps  are 
given  for  facilitating  interpolation.  For  this  purpose  auxiliary  tables  of  propor- 
tional parts  are  given  on  the  margins  of  the  pages  throughout  the  logarithmic 
portions  of  the  book.  In  general,  the  differences  in  the  table  of  the  natural  trigono- 
metric functions  are  so  small,  that  the  interpolations  can  easily  be  made  without  the 
use  of  the  tables  of  proportional  parts.  They  are,  therefore,  omitted  in  this  table 
and  also  in  the  table  of  squares,  etc.,  where  interpolations  are  seldom  necessary. 

Throughout  the  greater  part  of  the  book,  every  tenth  number  is  enclosed  by 
parallel  lines  and  a  space  is  left  between  every  three  numbers.  This  is  for  the  pur- 
pose of  giving  the  pages  a  pleasing  appearance  and  of  enabling  the  values  to  be 
readily  found.  In  the  trigonometric  tables,  it  has  been  the  aim  to  secure  a  sym- 
metrical arrangement,  so  that  in  reading  from  the  bottom  of  the  page,  the  order  is 
the  same  as  that  from  the  top. 

The  auxiliaries  S  and  T  are  given  at  the  bottoms  of  the  pages  in  the  table  of 
the  logarithms  of  numbers.  They  are  always  used  in  connection  with  the  logarithms 
of  numbers,  and,  consequently,  this  arrangement  is  more  convenient  than  having 
them  in  a  separate  table.  Their  arithmetical  complements  C  S  and  C  T  are  given  in 
the  table  of  the  logarithms  of  the  trigonometric  functions. 

The  tables  of  addition  and  subtraction  logarithms  are  based  on  those  of  Zech. 
The  argument  is  always  obtained  by  subtracting  the  smaller  from  the  larger  loga- 
rithm. In  addition  the  function  is  always  added,  and  in  subtraction  it  is  always 
subtracted  from  the  larger  logarithm.  On  account  of  these  uniform  ways  of  pro- 
ceeding, these  tables  are  more  convenient  than  the  usual  Gaussian  tables. 

Great  care  has  been  taken  to  secure  accuracy  in  the  tables.  The  proofs  have 
been  read  very  carefully.  Excepting  in  the  introduction  and  in  the  table  of  con- 
stants, only  four  errors  have  been  detected  in  the  first  edition.  The  correct  values 
of  the  mantissae  of  the  logarithms  of  5360  and  5489  are  .72916  and  .73949;  the  square 
of  881  is  776161;  the  cube  root  of  1008  is  10.0266.  All  known  errors  of  the  first  edition 
have  been  corrected  *in  this  one. 

Acknowledgment  is  due  to  Mr.  Taka  Kawada,  student  in  the  University,  for 
much  careful  assistance  in  reading  the  proofs  of  both  editions,  and  to  Professor 
W.  W.  Campbell,  Astronomer  in  the  Lick  Observatory,  for  valuable  suggestions  and 
for  permission  to  use  the  collection  of  formulas  resulting  from  the  method  of  least 
squares,  contained  in  his  Practical  Astronomy. 

W.  J.  IIUSSEY. 
ANN  ARBOR,  MICH.,  March  12,  1892 


CONTENTS. 


PAGE. 

INTRODUCTION, vii 

Eules  for  the  use  of  Logarithms, vii 

Common  Logarithms, vii 

Mantissae  of  Common  Logarithms, viii 

Characteristics  of  Common  Logarithms, viii 

The  Arithmetical  Complement  of  a  Logarithm, viii 

Explanations  of  the  Tables, ix 

THE  LOGARITHMS  OF  NUMBERS,        .      • ,       .       ,       1-22 

Common  Logarithms  of  Numbers  from  1  to  1000,     .        .        . .      .       .2-3 

"  "  "  "      100  to  10000,       .        .        .        .        4-21 

Proportional  Parts, 4-21 

Auxiliaries  S  and  T, .       .         2-21 

Natural  Logarithms  of  Whole  Numbers  from  1-200      ....  22 

ADDITION  AND  SUBTRACTION  LOGARITHMS, 23-  36 

Proportional  Parts 24-36 

Explanatory  Formulas, 24-  36 

LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS,        •  37-106 

For  the  First  and  Last  Degrees  of  all  Quadrants,        .        .        .        .37-46 

Proportional  Parts, •    ,   .        .       .        .38-49 

For  the  next  Five  Degrees  for  all  Quadrants,        .        .        .       .       .        50-60 

Proportional  Parts, 50-60 

For  the  entire  Circumference  from  Minute  to  Minute,        .       .       .       61-106 

Differences  and  Proportional  Parts, 62-106 

Auxiliaries  C  S  and  C  T, 62-64 

NATURAL  TRIGONOMETRIC  FUNCTIONS, *    .       .       .    107-130 

SQUARES,  CUBES,  SQUARE  ROOTS  AND  CUBE  BOOTS, 131-139 

FACTORS  OF  COMPUTING  PROBABLE  ERRORS,   .......          140 

TRIGONOMETRIC  FORMULAS, 141-142 

General  Formulas, 141 

Formulas  Relating  to  Plane  Triangles, 141 

"  "  "    Spherical  Triangles, 142 

FORMULAS  RESULTING  FROM  THE  METHOD  OF  LEAST  SQUARES,  .       .       143-145 
,    Formulas  for  the  Combination  of  Observations  and  the  Determina- 
tion of  Probable  Errors, 143 

CONSTANTS, 146-148 

Mathematical  and  Astronomical  Constants, 146 

Comparison  of  Linear  Measures, 146 

Dimensions  of  the  Earth  according  to  Bessel, 147 


VI  CONTENTS. 

Dimensions  of  the  Earth  according  to  Clarke,        .... 
Constants  for  Reducing  to  and  from  C.  G.  S.  System  of  Measures, 

Length, 

Area, .        .        . 

Volume, 

Mass, 

Velocity  and  Acceleration, 

Density, 

Force  in  Absolute  Measure, 

Work  and  Energy  in  Absolute  Measure,       .... 

Work  in  Gravitation  Measure, 

Rate  of  Working  in  Absolute  Measure,        .... 

"     "           "          "    Gravitation  Measure, 
Other  Physical  Constants, 


INTRODUCTION. 


Logarithms  are  used  in  lengthy  numerical  calculations  to  diminish  the  labor 
of  multiplication,  division,  involution  and  evolution,  by  respectively  substituting 
for  them  the  operations  of  addition,  subtraction,  multiplication  and  division. 

The  rules  for  their  use  are  as  follows  : 

The  logarithm  of  a  product  is  equal  to  the  sum  of  the  logarithms  of  its 
factors. 

The  logarithm  of  a  quotient  is  equal  to  the  logarithm  of  the  dividend,  minus 
the  logarithm  of  the  divisor. 

The  logarithm  of  any  power  of  a  number  is  equal  to  the  logarithm  of  the 
number  multiplied  by  the  index  of  the  power. 

The  logarithm  of  any  root  of  a  number  is  equal  to  the  logarithm  of  the  number 
divided  by  the  index  of  the  root. 

Or,  expressed  in  formulas, 


log  A»        =n  log  A,  log  PA  = 

These  rules  are  true  for  all  systems  of  logarithms.  The  Common  Logarithms 
are  the  only  ones  used  in  numerical  calculations  and  in  the  following  pages  they  are 
always  meant  unless  the  contrary  is  stated. 

The  common  logarithm  of  a  given  number  is  the  index  of  that  power  of  10 
which  is  equal  to  the  number.  Thus,  2  is  the  logarithm  of  100,  because  102  =  100; 
this  equation  is  usually  written  log  100  =  2.  10  is*  the  base  of  the  system.  A  system 
of  logarithms  comprises  the  logarithms  of  all  positive  numbers  to  a  given  base. 

From  the  definition  of  common  logarithms  it  follows,  that 

logl       =      0,  log  0.1       =  —  1, 

log  10     =  +  1,  log  0.01     =  —  2, 

log  100   =  +  2,  log  0.001    =  —  3, 

log  1000  =  +  3,  log  0.0001  =  —  4, 

etc.,  etc., 

from  which  it  is  evident,  that  logarithms  are,  in  general,  not  integers.    Thus,  the 
logarithm  of  a  number  between 

0.01  and       0.1  is  —  2  +  a  fraction, 

0.1    and       1  is  —  l-\-a  fraction, 

1      and     10  is      0  +  a  fraction, 

10      and    100  is      1  +  a  fraction, 

100      and  1000  is      2  -f  a  fraction. 

The  fractional  part  of  a  logarithm  is  usually  expressed  decimally  and  is  so 
taken  as  to  be  positive.    It  is  then  called  the  mantissa,  and  the  integral  part  is  called 
the  characteristic. 
vii 


Vlll  INTRODUCTION. 

Changing  the  decimal  point  in  a  number  is  equivalent  to  multiplying  or 
dividing  it  by  an  integral  power  of  10;  consequently,  the  logarithms  of  numbers 
which  are  the  same,  excepting  the  position  of  the  decimal  point,  differ  by  integers. 
Thus  the  logarithm  of  389.4  is  2.59040,  and  since  38940  =  100  X  389.4,  the  first  rule  for 
the  use  of  logarithms  gives 

log  38940  =  log  100  + log  389.4 

=      2      +2.59040  =  4.59040. 
Similarly, 

log  3.8940  =  log  .01  +  log  389.4 

=    —  2    +2.59040  =  0.59040. 
Hence, 

The  mantissas  of  the  logarithms  of  all  numbers  composed  of  the  same  figures 
in  the  same  order,  are  the  same. 

The  value  of  the  characteristic  depends  upon  the  position  of  the  decimal 
point  in  the  number.  An  inspection  of  the  above  table  shows,  that 

The  characteristic  of  the  logarithm  of  a  number,  partly  or  wholly  integral,  is 
zero  or  positive,  and  one  less  than  the  number  of  figures  in  the  integral  portion  ; 

The  characteristic  of  the  logarithm  of  a  pure  decimal  is  negative,  and  one 
more  than  the  number  of  ciphers  preceding  the  first  significant  figure. 

Examples:  The  mantissae  of  the  logarithms  of  349600,  3496,  3.496,  .003496  are 
the  same,  being  .54357;  their  characteristics  are +5, +3,0  and— 3,  respectively. 
Thus,  log  .003496  =  3.54357,  the  minus  sign  being  placed  over  the  characteristic  to 
indicate  that  it  only  is  negative. 

The  rule  given  above  for  determining  the  characteristic  of  the  logarithm  of  a 
pure  decimal  is  strictly  correct,  and  so  also  is  the  manner  of  writing  the  negative 
characteristic.  In  computing,  however,  it  is  not  desirable  to  use  the  characteristics 
in  the  manner  indicated.  It  is  preferable  to  add  10  to  logarithms  having  negative 
characteristics  and  to  allow  for  the  increase  by  a  proper  interpretation  of  the  results. 
When  so  increased  the  characteristics  may,  in  all  operations,  except  in  some  cases 
in  the  extraction  of  roots,  be  treated  as  if  they  were  positive.  When  written  in  this 
manner,  the  rule  for  their  determination  is  as  follows: 

The  characteristic  of  the  logarithm  of  a  pure  deimal  is  9,  diminished  by  the 
number  of  ciphers  preceding  the  first  significant  figure. 

Examples:  The  characteristics  of  the  logarithms  of  .8437,  .02804,  .000105  and 
.000009207  are  respectively  9,  8,  6  and  4. 

The  logarithmic  trigonometric  functions,  and  the  logarithms  of  constants  less 
than  unity  contained  in  these  tables,  have  had  their  characteristics  increased  bv  10. 

In  finding  the  logarithm  of  a  root  an  apparent  difficulty  arises  when  the  char- 
acteristic is  negative  and  is  not  a  multiple  of  the  index  of  the  root.  The  difficulty 
disappears  by  increasing  the  characteristic  negatively  by  the  smallest  number  which 
will  make  it  such  a  multiple  and  by  increasing  the  mantissa  positively  by  the  same 
number.  Thus,  the  logarithm  of  .003392  is  3.53046.  The  logarithm  of  its  square 
root  is  obtained  by  writing  its  logarithm  in  the  form— 4  + 1.53046  and  dividing  by  2, 
the  index  of  the  root.  This  gives  —  2  +  .76523,  or  2.76523,  or  8.76523. 

A  better  way  of  proceeding  is  to  add  10  times  the  index  of  the  root  to  the  log- 
arithm and  then  divide  by  the  index  of  the  root.  Thus,  in  tluM-xamplc  irivm.  adding 
20  to  the  logarithm  of  .003392  and  dividing  by  2,  gives  8.7(>523,  which  is  tin*  logarithm 
of  the  square  root.  By  adding  30  and  dividing  by  3,  the  logarithm  of  the  cube  root 
is  obtained.  Tin-  logarithm  of  the  cube  root  of  .003392  is  9.17' 

The  arithmiit'ul  r,,mplement  of  a  logarithm  is  the  dinVrence  obtained  i.y 
subtracting  it  from  0,  or  from  10,  if  it  is  desired  to  avoid  negative  characteristics. 


INTRODUCTION.  ix 

It  is  easily  obtained  by  subtracting  each  figure  of  the  logarithm,  except  the  last 
significant  one,  from  9;  the  last  significant  figure  must  be  subtracted  from  10. 
Thus,  log  2763  =  3.44138,  and  its  arithmetical  complement  is  6.55862.  It  is  to  be 
noticed,  that  the  logarithm  of  the  reciprocal  of  a  number,  is  the  arithmetical  com- 
plement of  the  logarithm  of  the  number;  for  example,  log  ^-^  =  6.55862. 

Since  the  sine  and  cosecant,  cosine  and  secant,  tangent  and  cotangent  are 
reciprocals,  their  logarithms  ara  arithmetical  complements.  Thus,  log  sin  22°  18' 
24"=  9.57928,  and  log  cosec  22°  18'  24"=  0.42072 ;  log  cos  22°  18'  24"=  9.96622,  and 
log  sec  22°  18'  24" =0.03378;  log  tan  22°  18'  24"=  9.61306,  and  log  cot  22°  18'  24"= 
0.38694. 

A  dash  printed  over  a  terminal  5  indicates  that  the  true  value  is  less  than  5. 
For  example  the  logarithm  of  59903  to  seven  decimal  places  is  4.7774486;  to  five 
decimal  places  this  is  written  4.77745.  If  only  four  decimal  places  are  required  in  a 
computation,  the  5  is  neglected.  Thus,  the  above  logarithm  is  written  4.7774. 

When  a  dash  is  not  printed  over  a  terminal  5,  and  only  four  decimal  places 
are  required,  the  fourth  decimal  figure  is  increased  by  one  and  the  5  neglected.  For 
example,  the  logarithm  of  7671  to  five  decimal  places  is  3.88485;  to  four  decimal 
places  this  is  written  3.8849. 

TABLE  I 

Pages  2-3  contain  the  mantissae  of  the  logarithms  of  all  numbers  of  one,  two 
and  three  figures;  the  characteristics  are  determined  by  the  rules  previously  given. 
If  the  number  has  one  or  two  figures,  it  is  given  in  the  first  column,  headed  !N",  and 
the  mantissa  of  its  logarithm  is  directly  opposite  it  in  the  second  column,  headed  L. 
Thus,  log  3=0.47712,  log  24  =  1.38021,  log  .067  =  8.82607.  If  the  number  has  three 
figures,  the  first  two  are  given  in  the  first  column  and  the  third  in  the  horizontal 
row  at  the  top  or  bottom  of  the  page,  and  the  mantissa  of  its  logarithm  is  at  the 
intersection  of  the  line  containing  the  first  two  figures  and  the  column  containing 
the  third.  Thus,  log  184  =  2,26482,  log  89.1  =  1.94988,  log  9.37  =  0.97174. 

Pages  4-21  contain  the  mantissae  of  the  logarithms  of  numbers  from  100  to 
10009.  The  arrangement  is  similar  to  that  just  described.  The  first  three  figures  of 
the  number  are  given  in  the  first  column  and  the  fourth  in  the  horizontal  row  at 
the  top  or  bottom  of  the  page.  The  last  three  figures  of  the  mantissae  are  given  in 
the  columns  headed  0,  1,  2,  3,  4,  5,  6,  7,  8,  9,  and  the  first  two.  at  intervals,  in  the 
second  column  under  L.  When  the  first  two  are  not  given  in  any  line,  they  are  to 
be  taken  from  the  first  line  above  containing  them,  except,  when  the  last  three  are 
preceded  by  a  *,  in  which  case  they  are  to  be  taken  from  the  next  line.  Thus,  (p.  13) 
log  5764  =  3.76072,  log  58.35  =  1.76604,  log  .5889  =  9.77004. 

When  the  number  has  more  than  four  figures,  its  logarithm  is  found  by  inter- 
polation. For  small  differences,  it  is  assumed,  that  differences  between  numbers 
are  proportional  to  the  differences  between  their  logarithms.  For  example,  required 
the  logarithm  of  168.342.  The  number  has  three  orders  of  integers,  hence  the 
characteristic  is  2.  Disregarding  the  decimal  point,  the  number  is  168342.  The 
round  numbers,  having  four  significant  figures,  next  smaller  and  next  greater  than 
this,  are  168300  and  168400,  and  their  mantissae  are  (p.  5)  .22608  and  .22634.  These 
numbers  differ  by  100,  their  mantissae,  by  26.  26,  being  the  difference  between  two 
successive  values  in  the  table,  is  the  tabular  difference.  168342  is  42  greater  than 
168300,  hence  its  mantissa  is  TVg-  of  26  (=  11,  to  the  nearest  integer,)  greater  than  that 
of  168300.  Therefore,  log  168.342  =  2.22619.  Similarly,  log  39.6427  =  1.59816. 


T  INTRODUCTION. 

To  facilitate  interpolation,  the  tenths  of  the  tabular  differences  are  given 
under  P  P,  (proportional  parts).    Thus,  from  the  proportional  table  for  26,  (p.  5), 
the  proportional  part  for  4   —  10.4 
&  "  "      "     2  =     .52 

Therefore,  "      "   42  =  10.92, 

or  11,  to  the  nearest  integer,  which  agrees  with  the  value  above. 

By  reversing  these  operations,  the  number  corresponding  to  a  given  logarithm 
may  be  found.  For  example,  find  the  number  of  which  1.47384  is  the  logarithm. 
The  next  smaller  mantissa  (p.  7)  is  .47378.  It  corresponds  to  the  number  2977.  The 
difference  between  it  and  the  next  greater  mantissa,  .47392,  is  14,  while  the  differ- 
ence between  it  and  the  given  mantissa  is  6.  The  figures  following  2977  are  obtained 
by  dividing  6  by  14,  giving  43.  Hence,  the  number  is  29.7743.  The  interpolation  is 
facilitated  by  using  the  proportional  table  for  14.  In  it,  5.6  is  the  value  next  smaller 
than  the  given  difference  6;  4,  the  fifth  figure  of  the  number,  corresponds  to  5.6. 
The  difference  between  6  and  5.6  is  .4,  which  becomes  4.0  by  removing  the  decimal 
point  one  place  to  the  right.  Corresponding  to  4.0,  the  nearest  value  is  3,  this  is  the 
sixth  figure  of  the  number.  The  interpolations,  where  proportional  parts  are  given, 
should  be  made  mentally,  the  results  only  being  written. 

The  logarithmic  sines  and  tangents  of  small  angles  may  be  found  by  means  of 
the  values  of  S  and  T,  given  at  the  bottoms  of  the  pages.  The  formulas  for  their  use 
are  as  follows: 

log  sin  =  log  arc  +  S, 
log  tan  =  log  arc  -f-  T, 

the  angle  being  expressed  in  seconds  of  arc.    The  value  of  S  or  T,  to  be  used  in  any 
case,  is  that  which  corresponds  to  the  angle. 
Example  1.    Find  log  sin  3".4785. 

log  3.4785  =  0.54139    p.  8. 
8  =  4.68557    p.  2. 
log  sin  3".4785=  5.22696. 

Example  2.    Find  log  tan  1°  U'  17".84  =log  tan  4457".84. 

log  4457//.84  =  3.64912    p.  10. 
T  =  4.68564    p.  10. 
log  tan  1°  W  17".84  =  8.33476. 

TABLE  II. 

When  the  logarithms  of  two  numbers  are  given  and  the  logarithm  of  their  sum 
or  difference  is  required,  it  may  be  found  by  using  the  addition  or  subtraction  table 
The  equations  at  the  bottoms  of  the  pages,  24-36  inclusive,  indicate  the  manner  of 
using  these  tables.  In  interpolating,  it  is  to  be  noticed  that  the  function  B  decreases 
as  the  argument  A  increases;  consequently,  the  proportional  parts  must  be  sub- 
tracted instead  of  added. 

Example  1.    Given,  log  a  =  0.98519  and  log  b  =  0.64834.    Required  log  (a  +  6). 

log  a  =  0.985 19 

log  6  =  0.64834 

A  =  log  a  -  log  6  =  0.33685 

B  =  0.16448    p.  24. 
log  (a  +  6)  =  log  a  +  B  =  1.14967. 

In  this  case  the  tabular  difference  is  31,  the  proportional  table  for  31  gives  28 
as  the  proportional  part  corresponding  to  85,  the  last  two  figures  of  A  ;  subtracting 


ItfT&ODtJCTlON.  Xl 

26  from  0.16474,  the  value  of  B  in  the  table  corresponding  to  a  value  of  A  =  0.33600, 
gives  0.16448.    This  is  the  value  of  B  corresponding  to  A  =  0.33685. 
Example  2.    Given,  log  a  and  log  6,  as  in  Example  1.    Required  log  (a-—  6). 
In  this  case  x  =  log  a  —log  6  is  >.3,  and,  as  above, 
A  =  log  a— log  6  =  0.33685 

B  =  0.26794    p.  29. 
log  (a  —  6)  =  log  a  —  B  =  0.71725. 

Example  3.    Given,  log  a  =  0.74346  and  log  b  =  0.59484.    Required  log  (a— 6). 
In  this  case  x  =  log  a— log  6  is  <.3,  and 

B  =  log  a— log  6  =  0.14862 

A  =  0.53790    p.  33. 
log  (a  —  6)  =  log  a — A  =  0.20556. 

TABLES  III  AND  IV. 

These  tables,  pp.  37-106,  contain  the  logarithms  of  the  trigonometric  functions. 
The  headings  of  the  pages  and  columns  indicate  what  they  contain.  The  degrees 
are  given  at  the  tops,  and  bottoms,  of  the  pages.  On  pp.  37-49,  the  minutes  and  each 
ten  seconds  are  given  in  columns  at  the  left  and  right,  headed  '  ",  and  the  odd 
seconds  are  given  in  a  horizontal  row  at  the  top  and  bottom  of  each  page.  On 
Pp.  50-106,  the  minutes  are  given  in  columns  at  the  left  and  right,  headed  '  ;  and  on 
pp.  50-60,  each  ten  seconds  is  given  in  a  horizontal  row  at  the  top  and  bottom  of  each 
page.  The  columns  of  minutes  on  the  left  read  downward;  the  horizontal  rows  at 
the  top,  from  left  to  right;  these  go  with  the  degrees  at  the  tops  of  the  pages.  The 
columns  of  minutes  at  the  right  and  the  horizontal  rows  at  the  bottom,  read  in  the 
opposite  directions,  and  go  with  the  degrees  at  the  bottoms  of  the  pages.  On  pp.  62- 
106,  the  tabular  differences  of  the  logarithmic  sines  and  cosines  are  given  in  the 
columns  headed  d  (difference),  and  those  of  the  logarithmic  tangents  and  cotangents 
in  the  columns  headed  c  d  (common  difference)* 
Example  1.  Find  log  sin  0°  37'  24".37. 

Page  44.  log  sin  0°  37'  24"     =  8.03659     Tabular  difference  =  19. 

proportional  part  for  3   =  5.7 

^  "  "     "    7=  1.33 

log  sin  0°  37'  24".37  =  8.03666. 

The  tabular  difference  is  19  and  the  proportional  table  for  19  (p.  45),  is  used  to 
facilitate  the  interpolation.  The  tabular  difference  is  obtained  by  subtracting  log 
sin  0°  37'  24'  =  8.03659  from  log  sin  0°  37'  25"  =  8.03678.  In  performing  this  subtrac- 
tion, only  the  final  figures  of  the  logarithms  need  be  used.  Thus,  in  this  case,  sub- 
tract 59  from  78.  The  interpolation  should  be  made  mentally  and  only  the  final 
result  written. 
Example  2.  Find  log  tan  0°  42'  17".48. 

Page  47.  log  tan  0°  42' 17"      =8.08992      Tabular  difference —17. 

proportional  part  for  .48  =          8.16 

log  tan  0°  42'  17".48  =  8.09000. 
Example  3.    Find  log  cos  0°  57'  19". 

This  is  given  without  interpolation  in  the  first  column  of  page  48,  the  first 
figures  being  given  at  the  top  of  the  column.    The  value  is  9.99994. 
Example  4.    Find  log  cos  89°  43'  26".4. 

Page  40.  log  cos  89°  43'  26"    =  7.68296     Tabular  difference  =  44. 

proportional  part  for  4  =         17.6 
log  cos  89°  43'  26".4  =  7.68278. 


Xli  INTRODUCTION. 

The  proportional  part  is  subtracted,  because  the  cosine,  here,  decreases  as  the 
angle  increases. 
Example  5.    Find  log  sin  3°  27'  44".  6. 

Page  54.  log  sin  3°  27'  40"    =  8.78083     Tabular  difference  =  35. 

proportional  part  for  4       =         14.0 
^  "  "     «        6  =  2.1 

log  sin  3°  27'  44".6  =  8.78099. 
Also  from  pages  54  and  55, 

log  cos  3°  27'  44".6  =  9.99920. 

log  tan  3°  27'  44".6  =  8.78178. 
Example  6.    Find  log  tan  8°  33'  17".4. 

Page  70.  log  tan  8°  33'  00"    =  9.17708     Tabular  difference  =  86 

proportional  part  for  10       =         14.3 
"     "    7       =       .   10.0 
•rV  "  "     "         4  =  .57 

log  tan  8°  33'  17".4  =  9.17733. 

Example  7.    Find  log  cot  56°  43'  24".7. 

Page  95.  log  cot  56°  43'  00"    =  9.81721     Tabular  difference  =  28. 

proportional  part  for  20       =  9.3 

"     "      4        =  1.9 

iV  "  "     "  7=  .33 

log  cot  56°  43'  24".7  =  9.81709. 

When  the  logarithm  of  a  trigonometric  function  is  given,  the  angle  may  be 
found  by  reversing  the  above  operations. 
Example  8.    Given,  log  tan  x  =  9.87258.    Find  x. 

In  the  column  of  logarithmic  tangents  on  page  98,  we  find  log  tan  36°  42'  = 
9.87238,  with  the  tabular  difference  26.    The  difference  between  this  logarithm  and 
the  given  one  is  20.    The  proportional  table  for  26  gives 
proportional  part  for  40      =  17.3 
"     "    6      =   2.6 

TV  "  "     "       2   =     .09 

consequently  "  "     "  46.2   =  19.99,  or  very  nearly  20. 

Hence  the  number  of  seconds  is  46.2,  and  the  required  angle  is  36°  42'  46".2. 

"When  a  very  small  angle  is  to  be  found  by  means  of  its  logarithmic  sine  or 
tangent,  and  accuracy  is  desired,  the  arithmetical  complement  of  S  or  T,  pp.  2-21, 
should  be  used.  These  are  given  in  the  columns  headed  C  S  and  C  T,  pp.  62-64. 
The  formulas  for  their  use  are  as  follows: 

log  arc  =  log  sin  +  C  S, 

log  arc  =  log  tan  -f-  C  T, 

the  angle  being  expressed  in  seconds  of  arc.    The  value  of  C  S  or  C  T  to  be  used  in 
any  case,  is  that  which  corresponds  to  the  angle. 

Example  9.    Given,  log  sin  x  =  6.82973.    Find  x. 

The  value  of  x,  (see  p.  62),  lies  between  0°  2'  and  0°  3',  or  between  120"  and 
180",  and,  corresponding  to  this, 

08  =  5.31443 


log  arc  =  2.  144  16. 

The  number  corresponding  to  the  logarithm  2.14416  is,  (p.  4),  139.368.    There- 
fore, x  =  139".368  =  0    2'  1U".308. 


INTRODUCTION. 


xiii 


It  is  sometimes  required  to  find  the  logarithm  of  one  trigonometric  function 
from  that  of  another,  without  requiring  the  angle.  To  facilitate  this,  special  pro- 
portional tables,  headed  with  the  tabular  differences  of  both  functions,  are  given, 
(pp.  71-106),  wherever  the  space  admits  it. 

Example  10.    Given,  log  tan  x  =  9.67644.    Required  log  cos  x. 

The  difference  between  the  given  logarithm  and  that  given  in  the  table, 
9.67622,  (see  p.  87,  opposite  25°  23'),  is  22.  The  tabular  differences  of  the  two 
logarithmic  functions  at  this  place  are  32  and  6.  In  the  proportional  table  for  s%,  22 
corresponds  to  4;  this,  subtracted  from  the  tabular  logarithmic  cosine  9.95591,  gives 
the  required  log  cos  x  =  9.95587. 

In  the  examples  already  given,  the  angles  have  all  been  less  than  90°.  The 
logarithms  of  trigonometric  functions  of  angles  greater  than  90°  may  be  obtained  by 
remembering  the  relations  given  in  the  following  table: 


Angle 

Sine 

Cosine 

Tangent 

Cotangent 

£C 

-[-sin  a? 

+cosa? 

+  tanoj 

+  cot  a; 

90°+# 

-f-cosa? 

—  sin  a; 

—  cot  x 

—  tana? 

180°  +0; 

—  since 

—  cos  a? 

-(-tanas 

+cot  x 

270°  +x 

—  cos  a? 

-f-sina? 

—  cot  x 

—  tana? 

For  angles  greater  than  90°,  the  degrees  are  given  at  the  tops  and  bottoms  of 
the  pages  in  smaller  type.  Where  these  have  been  obtained  from  the  acute  aftgle 
on  the  same  page,  by  adding  90°  or  270°,  they  are  preceded  by  a  *.  This  indicates 
tli  at  the  co-function  is  to  be  taken.  Otherwise,  the  direct  function  is  to  be  taken. 
The  algebraic  signs  of  the  functions,  as  indicated  by  the  above  table,  must  be 
attended  to. 

Example  11.    Find  log  sin  112°  15'  17". 

Page  84.  log  sin  112°  15'  00"  =  9.96640     Tabular  difference  =  6 

proportional  part  for  17"  =  2,  nearly, 

log  sin  112°  15'  17"  =  9.96638. 

From  the  same  page,  log  tan  202°  28'  34"  =  9.61671,  log  cos  202°  28' 34"== 
9.96569M,  log  cot  292°  18'  37"  =  9.61314,,. 

In  the  last  two  examples  the  „  following  the  logarithm  indicates  that  the 
trigonometric  function  is  negative.  This  is  the  usual  way  of  indicating  that  the 
number  corresponding  to  a  logarithm  is  negative. 

TABLE  V. 

Pages  108-130  contain  the  natural  trigonometric  functions  for  each  minute. 
The  arrangement  is  the  same  as  that  of  the  logarithms  of  the  trigonometric  func- 
tions, pp.  62-106,  except  that  differences  and  proportional  parts  are  not  given. 

TABLE  VI,  ETC. 

Pages  131-139  contain  the  squares,  cubes,  square  roots  and  cube  roots  of 
numbers  from  1  to  1020.  The  arrangement  of  this  table,  and  also  of  the  ones  which 
follow  it,  will  be  understood  by  inspecting  them. 


TABLE  OF  THE  COMMON 

LOGARITHMS  OF  NUMBERS 

WITH  THE  AUXILIARIES  S  AND  T. 


0—50 


N 

LO 

1   |   2 

3 

4 

5 

6 

7 

8 

9 

0 
i 

2 

3 

—  oo 

oo  ooo  30  103 

47  712   60  206 

69  897 

778i5 

84510 

90  309 

95424 

00000 

30103 
47712 

04139 

32  222 

49136 

07  918 
34242 
50515 

ii  394 
36  1  73 
51851 

14613 
38021 
53U8 

17609 
39794 
54407 

20412 
4i  497 
556jO 

23045 
43  136 
56820 

25  527 
44716 
57978 

27875' 
46240 
59  1  06 

4 
5 
6 

60206 

69897 
77815 

61  278 
70757 
78533 

62325 
71  600 

79239 

63  347 
72428 

79934 

64345 
73239 
80618 

65321 
74036 
Si  291 

66276 
74819 
81954 

67  210 

75587 
82  607 

68  124 

76343 
83251 

69  020 
77085 

83885 

7 
8 

9 
10 
ii 

12 
13 

84510 
90309 
95424 

85  126 
90849 
95904 

85733 
91381 

963/9 

86332 
91  908 
96848 

86923 
92428 
97313 

87  506 
92942 
97772 

88081 
93  450 
98227 

88649 
93952 
98677 

89  209 
94448 
99123 

89763 
94939 
99  564 

03743 

ooooo 

00432 

00860 

01  284 

oi  703 

02  119 

02531 

02  938 

03342 

04139 

07  918 

11394 

04^32 
08279 
ii  72? 

04922 
08636 
12057 

05  308 
08  991 
12385 

05  690 
09342 

12  7IO 

06070 
09691 
13033 

06  446 
10037 
13354 

06819 
10380 
13672 

07  188 
10  721 
13988 

07555 
11059 
14301 

14 

15 
16 

14613 
17609 

20412 

14922 

17898 
20683 

15229 
18  184 
20952 

15534 
18469 

21  219 

15836 
18752 
21  484 

16137 
19033 
21748 

16435 
19312 

22  Oil 

16732 

19590 
22  272 

17  026 
19866 
22531 

I73I9 

20  140 
22789 

i? 
18 

19 

20 

21 
22 
23 

23045 

25  527 
27875 

23300 
25768 
28  103 

23  553 
26007 
28330 

23805 
26245 
28556 

24055 
26482 
28  780 

24304 

26717 
29003 

24551 
26951 
29  226 

24797 
27184 
29447 

25042 
27416 
29667 

25285 
27646 
29885 

30103 

30320 

30535 

30750 

30963 

3i  175 

31387 

3i  597 

31  806 

32015 

32  222 
34242 
36173 

32428 

34439 
36361 

32634 
34635 
36549 

32838 
34830 
36736 

33041 
3502§ 
36922 

33244 

35218 

37  107 

33445 
354H 
37291 

33646 
35603 
37475 

33846 
35  793 
37658 

34044 

35984 
37840 

24 
25 
26 

38  O2  1 

39  794 
4i  497 

38  202 

39  9()7 
41664 

38382 
40140 
41  830 

38561 
40312 
41996 

38739 
40483 
42  1  60 

38917 
40654 

42325 

39°94 
40824 

42488 

39270 

40993 
42651 

39445 
41  162 
42813 

39  620 
41330 
42975 

2? 
28 

29 

30 

3i 

32 
33 

43  136 
44716 
46240 

43297 
448/1 
46389 

43457 
45025 
46538 

43616 

45  179 
46687 

43775 

45332 
46835 

43933 
45484 
46982 

44091 

45637 
47129 

44248 
45788 
47276 

44404 
45939 
47422 

4456o 
46090 
47  567 

47712 

47857 

48  ooi 

48144 

48287 

48430 

48572 

48  714 

48855 

48996 

49  136 

5051! 
51851 

49276 
50651 
5i983 

49415 
50786 

52114 

49554 
50  920 
52244 

49693 
5105! 
52375 

4983i 
51  188 
52504 

49969 
51322 
52634 

50  1  06 
51455 
52763 

50243 

51587 
52892 

50379 
51  720 
53020 

34 
35 
36 

53M8 
54407 
55630 

53275 
54531 

55  75i 

53403 
54  654 
55871 

53529 
54777 
55991 

53656 
54900 
56110 

53782 
55023 
56  229. 

53908 
55  145 
56348 

54033 
55267 

56467 

54158 
55388 
5658! 

54283 
55  509 
56703 

37 
38 
39 
40 

4i 
42 
43 

56820 
579/8 
59  1  06 

56937 
58092 

59218 

57054 
58206 

59329 

57I7I 
58320 

59439 

57287 
58433 
59550 

57403 
58546 
59660 

57519 
58659 

59770 

57634 

58771 
59879 

57749 
58883 
59988 

57864 
58995 
60097 

60206 

60314 

60423 

60531 

60638 

60  746 

60853 

60959 

61066 

61  172 

61  278 
62325 
63347 

61384 
62428 
63448 

6  1  490 

62531 
63548 

61  595 
62  634 

63649 

61  700 
62737 
63749 

61  805 
62839 
63849 

61909 
62941 
63949 

62  014 
63043 
64048 

62  118 

63144 
64147 

62  221 
63  246 

64  246 

44 
45 
46 

64345 
65321 
66276 

64  444 
65418 
66370 

64542 
655H 
66464 

64  640 
65  610 
66558 

64738 
65  706 
66652 

64836 
65  801 
66745 

64933 
65896 
66839 

65031 
65992 
66932 

65  128 
66087 
67025 

6522§ 
66l8l 
67II7 

47 

48 

49 
50 

67210 
68  124 
69020 

67302 
682i§ 
69  1  08 

67394 
68305 
69197 

67  486 

6839! 
6928§ 

67578 
68485 
69373 

67669 

68574 
69461 

67761 
68664 
69548 

67852 

68753 
69636 

67943 
68842 
69723 

68034 
68931 
69810 

69897 

'••)'£4 

70070 

70157 

70243 

70329 

70415 

70501 

705*6 

70672 

LO 

/ 

2 

8 

1 

5 

6 

7 

- 

0 

60'  =0°  i'   S   4-68557  T  4.68  557 

300"  =o°5'  S  4-68557  T  4.68558 

120  =0  2       4.68  557    4.68  557 

360  =06     4.68  557    4.68  558 

180  =o  3       4.68  557    4.68  557 

420  =07     4.68  557    4.68  558 

240  =•  o  4      4.68  557    4.68  558 

480  =08     4.68  557    4.68  558 

50—100 


N 

LO 

1 

2 

3  |   4 

5 

6 

7 

8 

9 

50 

52 
53 
54 
55 
,56 

57 
58 
59 
60 

61 
62 
63 
64 
65 
66 

67 
68- 

69 
70 

72 
73 
74 
75 
76 

77 
78 
79 

80 

Si 

82 
83 
84 

85 
86 

87 
88 
89 

90 

92 
93 
94 
95 
96 

97 
98 
99 
100 

69897 

69984 

70070 

70157    70243 

70329  [70415 

70  501 

70586 

70672 

70757 
71  600 
72428 

73239 
74036 
74819 

75587 
76343 
77085 

70842 
71684 
72509 
73320 
74H5 
74896 

75664 
76418 
77159 

70927 
71  767 

72  591 
73400 
74194 
74  974 

75  740 
76492 
77232 

71  OI2 
71  850 
72673 
73480 

-74  273 
75051 
758i5 
76  567 
77305 

71  096 
71933 
72754 
7356o 
74351 
75128 

75891 
76641 

77379 

71  181 

72  016 
72835 

73640 
74429 
75  205 

75967 
76716 
77452 

71265 
72099 
72916 

73719 

74507 
75282 

76  042 
76790 

77525 

71349 
72  181 
72997 

73799 
74586 
75358 
76118 
76864 
77597 

71433 
72  263 
73078 
73878 
74663 
75435 

76193 
76938 
77670 

72346 
73  159 
73957 
74741 
75  5H 
76268 
77012 
77743 

77815 

77887 

7796o 

78032 

78  104 

78176 

78247 

78319 

7839° 

78462 

78533 
79239 
79934 
80618 
81  291 
8i954 
82  607 
83251 
83  885 

78604 
79309 
80003 

80686 
81358 
82020 

-82672 

83315 
83948 

78675 

79379 
80072 

80754 
Si  425 
82086 

82737 
83378 
84  on 

78746 

79449 
So  140 

80821 
81  491 
82  151 
82802 
83442 
84073 

78817 
79  5i8 
80209 
So  889 
81558 
82217 

82866 
83506 
84136 

78888 
79588 
80277 
80956 
8  1  624 
82  282 
82930 

83569 
84  198 

78958 
79657 
80346 

81023 
8  1  690 
82347 
82995 
83632 
84  261 

79029 

79727 
80414 

8  1  090 

81757 
82413 

83059 
83696 

84323 

79099 
79796 
80482 
81  158 
81  823 
82478 

83123 

83759 
84386 

79169 
79865 
80550 

Si  224 

81889 
82543 
83187 
83822 
84448 

84510 

S45/2 

84  634 

84  696 

84757 

84819 

84880 

84942 

85003 

85065 

85  126 

85733 
86332 

86923 
87506 
88081 
88649 
89  209 
89763 

85187 
85  794 
86392 

86982 

87564 
88  138 

88  705 
89265 
89  818 

85248 

85854 
86451 

87040 
87622 
88  195 
88762 
89321 
89873 

85309 

86  510 
87099 
87  679 
88  252 

88818 
89376 
89927 

85370 
85974 
86570 

87157 
87737 
88309 

88874 
89432 
89982 

85431 
86034 
86629 
87216 

87795 
88366 

88930 

89487 
90037' 

85491 
86094 
86688 
87274 
87852 
88423 
88986 
89  542 
90091 

85552 
86153 
86  747 

87332 
87910 
88480 
89042 

89597 
90  146 

85612 
86213 
86806 
87390 
87967 
88536 
89098 

89653 
90  200 

85673 
86273 
86864 

87448 
88024 
88593 
89154 
89  708 
90255 

90309 

90363 

90417 

90472 

90526 

90  580 

90634 

90  687 

90741 

90795 

90849 
91381 
91  908 
92  428 
92942 
93450 
93952 
94448 
94939 

90902 

91  434 
91  960 

92  480 
92993 
93  5oo 
94002 
94498 
94  988 

90956 
91487 
92012 

92531 
93044 
93551 
94052 

94547 
95036 

91  009 
9i  540 
92065 

92583 
93095 
936oi 

94  ioi 
94596 
95085 

91  062 
9i  593 
92  117 
92634 
93  146 
93651 
94I5I 
94645 
95  134 

91  116 
91645 
92  169 
92686 
93  197 
93702 

94  201 
94694 
95  182 

91  169 
91  698 

92  221 
92737 

93  247 
93752 
94250 
94743 
95  231 

91  222 

91  751 
92273 

92788 

93  298 
93802 
94300 
94792 
95279 

91  275 
91  803 
92324 
92  840 
93349 
93852 

94349 
94841 
95328 

91  328 
91855 
92376 
92  891 
93399 
93902 

94399 
94890 
95376 

95424 

95472 

95521 

95  569 

95617 

95  665 

95  713 

95  761 

95809 

95856 

95904 
96379 
96848 

97313 
97772 
98  227 

98677 
99  123 
99  564 

95952 
96426 
96895 

97359 
97  818 
98  272 
98722 
99167 
99607 

95999 
96473 
96942 

97405 
97864 
98318 

98767 
99211 
99651 

96047 
96  520 
96988 

97451 
97909 
98363 
98  8n 
99255 
99695 

96095 
96567 
97035 
97497 
97955 
98408 

98856 
99300 
99  739 

96  142 
96614 
97081 

97543 
98000 

98453 
98900 
99344 

99  782 

96  190 
96661 
97128 

97589 
98046 
98498 

98945 
99388 
99  826 

96237 
96  708 
97174 

97635 
98091 

98543 
98989 
99432 
99870 

96284 

96755 
97  220 

97681 
98  137 

98588 

99034 
99476 
99913 

96332 
96802 
97267 

97727 
98  182 
98632 
99078 
99520 
99957 

oo  ooo 

00043 

00087 

oo  130 

00173 

oo  217 

00260 

00303 

oo  346 

00389 

N 

LO     1      2 

3      4 

5      6 

7 

8 

9 

540"  =0°  9'   S   4.68557  T  4.68  558 
600  =  o  10       4.68  557    4.68  558 
660  =  o  II       4.68  557    4.68  558 
720  =  o  12      4.68  557    4.68  558 

78o"  —  o°  13'  S  4.68557  T  4.68558 
840  =  o  14    4.68  557    4.68  558 
900  =  o  15    4.68  557    4.68  558 
960  =  o  16    4.68  557    4.68  558 

100—150 


N 

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1 

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3 

4 

5 

6  |  7 

8 

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100 

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130 

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217 

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4-4   4-3    4-2 

8.8   8.6   8,4 

13.2   12-9    12.6 

104 
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366 

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16.4  16.0   15.6 

116 

446 

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521 

558 

595 

633 

670 

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20.5  20.  o   19.5 
24.6  24.0   23.4 

118 
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225 
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262 
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28,7  28.0  27.3 
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11.4  ii.  I   10.8 

124 

125 
126 

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377 
726 
072 

412 
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447 
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140 

482 
830 
175 

517 
864 
209 

552 
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587 
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278 

621 
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15.2  14.8   14.4 
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22.8   22.2   21.6 

127 
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380 
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093 

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294 

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327 

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8 
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26.6   25.9   25.2 
394   29.6   28.8 

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156 

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287 

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133 

385 

418 

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516 

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10.5  10.2   9.9 

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840 

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14.0  13.6   13.2 

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258 

290 

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17.5  17.0   16.5 

136 

354 

386 

418 

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513 

545 

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609 

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21.0  20.4   19.8 

137 

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24.5  23.8   23.1 

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31.5  30.6   29.7 

140 

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534 

564 

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12.8   12.4   12.0 

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19.2  18.6   18.0 

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820 

850 

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938 

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148 

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114 

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231 

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25.6  24.8   24.0 

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28.8  27.9   27.0 

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667 

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725 

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960'  -o°  16'  8  4.68557  T  4.68558   1260"  =0°  21'  8  4.68557  T  4-68558 

1020  =0  17      4.68557      4.68558    1320  =0  22      4-68557      4.68558 

1080  =0  18    4.68557     4.68558   1380  =o  23    4-68557     4.68558 

1140  =o  19    4.68557     4.68558   1440  =°  24    4.68557     4-68558 

1200  =0  20      4.68557      4.68558    1500  =0  25      4-68557      4.68558 

150—200 


K 

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150 

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151 
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156 

157 
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21.6   20.8 

24.3  23.4 

170 

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15.4  14.7 

17.6  16.8 
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200 

30103 

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190 

211 

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255 

276 

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1500 
1560 
1620 
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1     4-68559 
1     4-68559 
1     4.68559 
7     4-68559 

200—250 


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320 
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212 
213 
214 

215 
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2 

8 

4 

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8 

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2040  =o  34    4.68  557     4.68  559 
2100  =o  35    4  68  557     4-68  559 
2160  =o  36    4.68557     4.68559 
2220  =0  37    4.68  557     4.68  559 

2280*  =0°  38'  S  4.68557  T  4.  <>8  559 
2340  =o  39    4-68557     4-68550 
2400  =o  40    4.68  557     4.68  559 
2460  =o  41     4.68556     4.68560 
2520  —o  42    4.68556     4.68560 

250—300 


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4.  68  560 
4.68  561 

300—350 


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12 

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301 
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120 
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145 
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P  P 

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3060  =  o  51     4.68  556    4.68  561 
3120  =o  52     4.68556    4.68561 
3180  =o  53     4.68556    4.68561 
3240  =  o  54     4.68  556    4.68  561 

3300*  =0°  55'  S  4.68556  T  4.68561 
3360  =  o  56     4.68  556    4.68  561 
3420  =  o  57     4.68  555     4.68  561 
3480  =  o  58     4.68  555     4.68  562 
3540  =  o  59     4.68  555     4.68  562 

350— 400 


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534 
616 

697 

779 
860 
941 

*022 
102 

I83 

542 
624 
705 

787 
868 

949 
*030 
in 
191 

550 
632 

713 

795 
876 

957 

*038 
119 
199 

558 
640 
722 
803 
884 
965 
4,046 
127 
207 

567 
648 
730 
811 
892 
973 
*054 
135 
215 

575 
656 
738 
819 
900 
981 
^062 

143 

223 

583 
665 
746 

827 
908 
989 
*070 
I5i 
231 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

320 
400 
480 
560 
640 
719 

799, 
878 

957 

328 
408 
488 
568 
648 
727 

807 
886 
965 

336 
416 
496 
5/6 
656 
735 
BiJ 
894 
973 

344 
424 

504 

584 
664 

743 
823 
902 
981 

352 
432 
512 

592 
672 

751 
830 
910 
989 

360 
440 
520 
600 
679 
759 
838 
918 
997 

368 
448 
528 
608 
687 
767 
846 
926 
*oo§ 

376 
456 
536 
616 
695 
77? 
854 
933 
*oi3 

384 
464 

544 
624 
703 
783 
862 
941 
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392 
472 
552 
632 
711 
791 
870 
949 

*028 

74036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

N 

L  0 

1 

2  |  3 

i 

5 

7 

8 

9 

'  P 

4980*  =  i°  23'  S  4-68553  T  4.68566 
5040  =  i  24     4.68  553     4.68  566 
5100  =  i  25     4.68  553    4.68  566 
5160  =  I  26     4.68  553     4.68  567 
5220  =  i  27     4.68  553     4.68  567 

5280'=  °28'  S  4.68553  T  4.68567 
5340  =   29     4.68  553     4.68  567 
5400  =   30     4.68  553     4-68  567 
5460  =   31     4.68  552     4.68  568 
5520  =   32     4.68  552     4.68  568 

13 


550—600 


N 

L  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P  P 

550 

74036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

8 
i   0.8 

2    1.6 

3   2.4 
4   3-2 
5   4-0 
6   4.8 

7   5-6 
8   6.4 
9   7-2 

7 
i   0.7 

2    1.4 

3   2.1 

4   2.8 

5   3-5 
6   4.2 

7   4-9 
8   5.6 
9   6.3 

55i 
552 
553 
554 
555 
556 

557 
558 
559 
560 

56i 
562 
563 
564 
565 
566 

567 
568 

569 

H5 
194 

273 

35i 
429 

507 
586 
663 
741 

123 
202 

280 

359 
437 
515 
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671 
749 

131 

2IO 

288 

367 

445 
523 
60  1 
679 

757 

139 
218 
296 

374 
453 
531 
609 
687 
764 

U7 

225 

304 
382 
461 

539 
617 
695 
772 

155 
233 
312 

390 
468 

547 
624 
702 
780 

162 
241 
320 

398 
476 

554 
632 
710 

788 

170 
249 
327 
406 
484 
562 
640 
718 
796 

178 
257 
335 
414 
492 
570 
648 
726 
833, 

1  86 
265 
343 
421 
500 
578 
656 

733 
811 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

896 
974 

75051 
128 
205 

282 

358 
435 
5ii 

904 
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059 
136 
213 
289 

366 
442 
519 

912 
989 
066 

143 

220 
297 

374 
450 
526 

920 

997 
074 

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228 

305 

38i 
458 
534 

927 

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082 

159 
236 
312 
389 
465 
542 

935 
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089 

1  66 

243 
320 

397 
473 
549 

943 

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097 

174 
251 

328 

404 
481 

557 

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105 
182 
259 

335 
412 
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958 

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H3 
189 
266 

343 
420 
496 
572 

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120 

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274 
351 

427 

504 

580 

570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 

57i 

572 
573 
574 
575 
576 

577 
578 
579 

580 

581 
582 
583 
584 
585 
586 

587 

588 

589 
590 

59i 
592 
593 
594 
595 
596 

597 
598 
599 
600 

664 
740 

8i5 
891 

967 

76042 

118 

193 

268 

671 
747 
823 
899 

974 
050 

125 
200 

275 

679 
755 
831 
906 
982 
057 

133 
208 

283 

686 
762 
838 
914 
989 
065 
140 

215 
290 

694 
770 
846 
921 

997 
072 

148 
223 

298 

702 

778 
853 
929 

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080 

155 
230 
305 

709 

785 
861 

937 
*°12 
087 

163 
238 
313 

717 
793 
868 

944 

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095 
170 

245 
320 

724 
800 

876 

952 

*027 

103 

178 

253 

328 

732 
808 
884 

959 
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no 

185 
260 
335 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

418 
492 

567 
641 
716 
790 
864 
938 
77012 

425 
500 

574 

649 
723 
797 
871 
945 
019 

433 
507 

582 

656 
730 

805 
879 

953 
026 

440 
515 

589 
664 

738 
812 

886 
960 
034 

448 
522 
597 
671 
745 
819 

893 
967 
041 

455 
530 
604 

678 

753 
827 

901 

975 
048 

462 

537 
612 

686 
760 
834 
908 
982 
056 

470 

545 
619 

693 
768 
842 
916 
989 
063 

477 
552 
626 

701 

775 
849 

923 
997 
070 

485 
559 
634 

708 
782 
856 

930 
*oo4 
078 

085 

093 

100 

107 

H5 

122 

129 

137 

144 

151 

159 
232 

305 

379 
452 

525 

597 
670 

743 

1  66 
240 
313 

386 

459 

532 

605 
677 
7§o 

173 
247 
320 

393 
466 

539 
612 
685 

757 

181 
254 
327 
401 
474 
546 
619 
692 
764 

188 
262 
335 
408 
481 
554 
627 
699 
772 

195 
269 
342 

415 
488 
561 

634 
706 

779 

203 
276 
349 
422 

495 
568 

641 

7H 
786 

2IO 

283 

357 
430 
503 
576 
648 
721 
793 

217 
291 

364 

437 
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583 
656 
728 
801 

225 
298 
37i 
444 
517 
59° 
663 

735 
808 

8i5 

822 

830 

.837 

844 

851 

859 

866 

873 

880 

N 

L  0  |  1    2  |  3 

4 

5 

6  . 

7 

8 

9 

P  P 

5460"  =1°  31'  S  4.68552  T  4.68568 
5520  =i  32    4.68552    4.68568 
5580  =i  33    4.68552     4.68568 
5640  =i  34    4.68552     4.68568 
5700  =i  35    4.68552     4.68569 

5760"  =1°  36'  S  4.68552  T  4.68569 
5820  =i  37    4.68552     4.68569 
5880  =i  38     4.68552     4.68569 
5940  =i  39    4-68551     4.68569 
6000  =i  40    4.68551     4.68570 

14 


600—650 


N 

LO 

1 

2 

3 

4 

5 

6 

7 

8 

9 

p 

P 

600 

77815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

601 
602 
603 

604 

605 
606 

607 
608 
609 

887 
960 
78  032. 
104 
176 
247 

3i9 
390 
462 

89! 
967 
039 
in 
183 
254 
326 
398 
469 

902 

974 
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118 
190 
262 

333 
405 
4/6 

909 
981 
053 
125 

'97 
269 

340 
412 

483 

916 
988 
06  1 

132 
204 
276 

347 
419 

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924 
996 
068 
140 

211 
283 

355 
426 

497 

93i 
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075 

147 
219 
290 
362 
433 
504 

938 
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082 

154 
226 

297 

369 
440 

512 

945 
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089 

161 
233 
305 
376 
447 
519 

952 
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097 

1  68 
240 
312 

383 
455 
526 

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2 

3 

8 
0.8 
1.6 
2.4 

610 

533 

540 

547 

554 

56i 

569 

576 

583 

590 

597 

4 

5 

3-2 
4.0 

611 
612 
613 
614 
615 
616 

617 
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619 

604 
675 
746 

817 
888 
958 
79029 
099 
169 

611 
682 
753 
824 
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965 
036 
106 
176 

618 
689 
760 

831 
902 
972 

043 
H3 
183 

625 
696 
767 
838 
909 
979 
050 
1  20 
190 

633 
704 

774 

845 
916 
986 

057 
127 

J97 

640 
711 
78i 
852 
923 
993 
064 
134 
204 

647 
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789 

859 
930 
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071 
141 

211 

654 
725 
796 
866 
937 
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078 
148 
218 

66  1 
732 
803 

873 
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085 

155 
225 

668 

739 
810 

880 
951 

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092 
162 
232 

6 

8 
9 

4.8 
5-6 
6.4 
7-2 

620 

239 

246 

253 

260 

267 

274 

28l 

288 

295 

302 

621 
622 
623 
624 
625 
626 

627 
628 
629 

309 
379 

449 
5i8 
588 
657 

727 
796 
865 

316 
386 
456 

525 
595 
664 

734 
803 
872 

323 
393 
463 
532 
602 
671 
741 
810 
879 

330 
400 
470 

539 
609 
678 

748 
817 
886 

337 
407 

477 
546 
616 

685 

754 
824 

893 

344 
414 

484 

553 
623 
692 
761 

831 
900 

351 
421 
491 

560 
630 
699 
768 

837 
906 

-358 
428 
498 

567 
637 
706 

775 
844 
913 

365 
435 
505 
574 
644 
713 
782 
851 
920 

372 
442 
511 
581 
650 
720 

789 

858 

927 

i 

2 

3 
4 
5 
6 

8 
9 

7 
0.7 
1.4 

2.1 

2.8 

3-5 
4.2 

4-9 
5-6 
6.3 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989 

996 

631 
632 
633 
634 
635 
636 

637 
638 

639 

80003 
072 
140 
209 
277 
346 
414 
482 
550 

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079 
147 
216 
284 
353 
421 
489 
557_ 

017 
085 
154 
223 
291 
359 
428 
496 
564 

024 
092 
161 
229 
298 
366 

434 

502 
570 

030 
099 
1  68 
236 
305 
373 
441 
509 
577 

037 
1  06 

175 

243 
312 
380 

448 
516 

584 

044 
H3 
182 

250 
3i8 
387 
455 
523 
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051 

120 

188 

257 
325 
393 
462 
530 
598 

058 
127 
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264 
332 
400 

468 
536 
604 

065 

134 
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271 

339 
407 

475 
543 
611 

i 

2 

6 

0.6 

1.2 

i  8 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

4 

2.4 

641 
642 
643 
644 

645 
646 

647 
648 
649 

686 

754 
821 

889 
956 
8  1  023 

090 
158 
224 

693 
760 

828 

895 
963 
030 

097 
164 
231 

699 
767 
835 
902 
969 
037 
104 

I7i 

238 

706 

774 
841 

909 
976 
043 

in 

178 
245 

713 

78i 
848 

916 
983 
050 

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184 
251 

720 
787 
855 
922 
990 
057 
124 
191 
258 

726 

794 
862 

929 
996 
064 

131 

198 
265 

733 
801 
868 

936 
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070 

137 
204 
271 

740 
808 
875 
943 
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077 
144 

211 

278 

747 
814 
882 

949 
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151 
218 

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6 

8 
9 

3-0 
3.6 
4-2 
4.8 
5-4 

650 

291 

298 

305 

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3i8 

325 

331 

338 

345 

351 

N 

L  0 

1 

2 

3 

4 

5 

6 

7 

8 

0 

I' 

P 

6000' 
6060 

6120 
6180 
6240 

=  i°4o' 
=  I  41 
=  I  42 
=  i  43 
=  I  44 

S 

4-f>«  : 
4.68  £ 
4.685 
4.685 
4-68  5 

5i  'J 
51 

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51 
51 

1  4-68 
4.68 
4.68 
4.68 
4.68 

570 
570 
570 
570 
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6. 
6, 
65 

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,60  = 

.20  = 
80  = 
40  = 

:  i°45 

:  I  46 
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:  I  48 
:  I  49 

'   S 

4-68  551 
4-68  551 
4-68  550 
4-68  550 
4-68  550 

T  4.68  571 
4-68  571 
4-68  572 
4-68  572 
4-68  572 

15 


650—700 


N 

L  0 

1 

2 

3    4 

5 

6 

7 

8  |  9 

P  P 

650 

651 
652 

653 
654 
655 
656 

657 
658 
659 
660 

661 
662 
663 
664 
665 
666 
667 
668 
669 

670 

671 
672 
673 
674 
675 
676 

677 
6/8 
679 

680 

81  291 

298 

305 

3u 

3i8 

325 

33i 

338 

345 

35i 

358 
425 
491 

558 
624 
690 

757 
823 
889 

36! 
43i 
498 

564 
631 
697 

763 
829 
895 

37i 
438 
505 
571 
637 
704 

770 

836 
902 

378 
445 
5ii 
578 
644 
710 

776 
842 
908 

38§ 
45i 

518 

584 
651 
717 

783 
849 

9i5 

39i 

458 
525 

59i 
657 
723 
790 
856 
921 

398 
465 
53i 

598 
664 
730 
796 
862 
928 

405 
47i 
538 
604 
671 
737 
803 
869 
935 

411 
478 
544 
611 
677 
743 
809 

875 
941 

418 
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551 

617 
684 
750 
816 
882 
948 

7 
i  0.7 

2   1.4 

3  2.1 

4  2.8 

5  3-5 
6  4.2 

7  4.9 

954 

961 

968 

974 

981 

987 

994 

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151 

217 
282 

347 

413 
478 

543 

027 
092 

158 
223 
289 
354 
419 
484 
549 

033 
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164 

230 

295 
360 

426 
491 
556 

040 

105 
171 

236 
302 
367 
432 
497 
562 

046 

112 

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243 

308 

373 

439 
504 
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053 
119 
184 

249 
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380 

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575 

060 
125 
191 

256 
321 

387 
45.2 
517 
582 

066 
132 
197 

263 

328 

393 

458 
523 

588 

073 
138 
204 
269 
334 
400 

465 
530 
595 

079 
145 

2IO 
276 

341 
406 

471 
536 
60  1 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

8  5.6 
o  6  ^ 

672 

737 
802 

866 
930 
995 

83059 
123 

187 

679 

743 
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872 
937 

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065 
129 
193 

685 
750 
814 

879 
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072 
136 
200 

692 
756 

821 

885 
950 
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078 
142 
206 

698 
763 
827 
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149 
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091 

155 
219 

711 
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840 

905 
969 
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097 
161 
225 

718 
782 
847 
911 
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104 

168 

232 

724 
789 
853 
918 
982 
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no 

174 

238 

730 

795 
860 

924 

988 

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117 

181 
245 

6 

i  0.6 

2   1.2 

3   1.8 
4  2.4 
5  3-0 
6  3.6 

7  4-2 
8  4.8 

9  5-4 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

681 
682 
683 
684 
685 
686 
687 
638 
689 

315 
3/8 
442 

506 
569 
632 
696 

759 
822 

321 
385 
448 
512 
575 
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702 

765 

828 

327 
391 

455 
5i8 
582 
645 
708 
77i 
835 

334 
398 
461 

525 
588 
651 

715 
778 
841 

340 
404 
467 
531 

594 

658 

721 
784 

847 

347 
410 

474 

537 
601 
664 
727 
790 
853 

353 
417 
480 

544 
607 
670 

734 
797 
860 

359 
423 
487 
550 
613 
677 
740 
803 
866 

366 
429 
493 
556 
620 
683 

746 
809 
872 

372 
436 
499 

563 
626 
689 

753 
816 
879 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

691 
692 
693 
694 

695 
696 

697 
698 
699 

948 
84011 
073 
136 
198 
261 

323 
386 
448 

954 
017 
080 
142 
205 
267 

330 

392 
454 

960 
023 

086 
148 

211 
273 
336 

398 
460 

967 
029 
092 

155 
217 
280 

342 
404 
466 

973 
036 
098 
161 
223 
286 

348 
410 
473 

979 
042 

105 
167 
230 

292 

354 
417 

479 

985 
048 
in 

173 
236 

298 

361 
423 

485 

992 
055 
H7 
1  80 
242 
305 
367 
429 
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998 
061 

123 
186 
248 
3ii 
373 
435 
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559 

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067 
130 
192 

255 
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379 
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504 

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510 

516  522 

528 

535 

541 

547 

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K 

L  0    1    2 

3    4 

5 

6 

7 

8 

9 

P  P 

6480"  --i°  48'  S  4.68550  T  4.68572 
6540  =i  49    4.68550     4.68572 
6600  =i  50    4.68550     4.68572 
6660  =i  51    4.68550     4.68573 
6720  =i  52    4.68550     4-68573 

6780"  =  °  53'  S  4.  68  550  T  4.68  573 
6840  -   54    4-68550     4.68573 
6900  =   55     4.68549     4.68574 
6960  =   56    4.68  549     4.68  574 
7020  =   57     4-68549     4.68574 

16 


700—750 


N 

L  0 

1 

2 

3 

4 

5    0 

7 

8    U 

p  r 

700 

84510 

5i6 

522 

528 

535- 

54i 

547 

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3 
4 

6 

8 
9 

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2 

3 
4 

6 

8 
9 

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2 

3 
4 

7 
0.7 
1.4 

2.1 

2.8 

3-5 
4.2 

4-9 
5-6 
6-3 

6 

0.6 
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1.8 
2.4 
3-0 
3-6 

4-2 

4.8 
5-4 

5 

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i.5 

2.0 
2.5 
3-0 
3-5 
4.0 

4-5 

701 
702 
703 
704 

705 

706 

707 
708 
709 

572 
634 
696 

757 
819 
880 

942 
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065 

5/8 
640 
702 

763 
825 
887 

948 
009 
071 

584 
646 
708 

770' 
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954 
016 

077 

590 
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7U 
776 
837 
899 
960 

022 

083 

597 
658 
720 
782 
844 
905 

967 
028 
089 

603 
66§ 
726 
788 
850 
911 

973 
034 
095 

609 
671 

733 

794 
856 

9J7 
979 
040 

IOI 

615 
677 
739 
800 
862 
924 

985 
046 
107 

621 
683 
745 
807 
868 
930 
991 
052 
114 

628 
689 
75i 

813 
874 
936 

997 
058-. 

120 

710 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

711 
712 
7i3 
7U 
7i5 
716 

7i7 
718 
719 

720 

187 
248 
309 
370 
431 
491 

552 
612 
673 

193 
254 
315 
376 
437 
497 
558 
618 
679 

199 
260 
321 
382 
443 
503 

564 
625 
685 

205 
266 
327 
388 

449 

509 
570 
631 
691 

211 

272 

333 
394 
455 
5i6 

576 
637 
697 

217 

278 

339 
400 
461 

522 

582 
643 
"03 

224 

285 
345 
406 
467 
528 
588 
649 
709 

230 
291 
352 
412 
473 
534 
594 
655 
715 

236 
297 
358 
418 

479 
540 
600 
661 
72i. 

242 
303 
364 
425 

485 
546 
606 
667 

727 

733 

739 

745 

75i 

757 

763 

769 

775 

781 

788 

721 
722 
723 
724 

725 
726 

727 
728 
729 

730 

73i 
732 
733 
734 
735 
736 

737 
738 
739 

740 

74i 

742 

743 
744 
745 
740 

747 
748 

749 

780 

794 

854 
914 

974 
86034 
094 

153 
213 

273 

800 
860 
920 
980 
040 

100 

159 
219 
279 

806 
866 
926 
986 
046 
1  06 

165 
225 
285 

812 
872 
932 

992 
052 

112 

171 
231 
29I 

818 
878 
938 

998 
058 
118 

177 

237 
297 

824 
884 
944 
*oo4 
064 
124 
183 
243 
303 

830 
890 
9§o 
*oio 
070 
130 
189 

249 
308 

836 
896 
956 

*OJ6 
076 
136 

J95 

255 
3i4 

842 
902 
962 

*-°22 
082 
141 

201 
261 
320 

848 
908 
968 

*02S 
088 
147 
207 
267 
326 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 

392 
451 
5io 

570 
629 
688 

747 
806 
864 

398 
457 
516 

576 
635 
694 

753 
812 
870 

404 

463 

522 

58i 
641 
700 

759 
817 
876 

410 

469 

528 

587 
646 

705 
764 
823 
882 

415 
475 
534 

593 
652 
711 

770 
.829 
888 

421 
481 
540 

599 

658 

717 
776 
835 
894 

427 
487 
546 
605 
664 
723 

782 
841 
900 

433 
493 
552 
611 
670 
729 
788 
847 
906 

439 
499 

558 

617 
676 
735 
794 
853 
911 

445 
504 
564 
623 
682 
74i 
800 
859 
9^7 

923 

929 

935 

941 

947 

953 

958 

964 

970 

9/6 

A 

982 
87040 
099 

157 
216 

274 
332 
39<> 
448 

988 
046 
105 
163 

221 
280 
338 
396 

454 

994 
052 
in 
169 

2?7 

286 

344 
402 
460 

999 
058 
116 

175 
233 
291 

349 
408 
466 

*oo5 
064 

122 

181 

239 

297 

355 
4U 
471 

*on 
070 
128 
1  86 
245 
303 
361 
419 
477 

*oi7 
075 
134 
192 
251 
309 
367 
425 
483 

*023 

081 
140 
198 
256 
315 
373 
431 
489 

*029 

087 

146 

204 
262 

320 

379 
437 
495 

*035 
093 
151 

210 

268 
326 

384 
442 
500 

8 
9 

506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

N 

L  0 

1 

3    4 

507 

8 

9 

1'  P 

,-,0*  -  i°56'  S 
7020  =  I  57 
7080  =  i  58 
7140  =  I  59 

720O  =  2   0 

4.68  549  T  4-68  574 
4.68  549     4.68  574 
4.68  549       *  575 
4-68  549       s  57! 
4.68549      •-•'575 

•-.,/  =  2*  r  s  4-68549  T  4-68575 

7320  =2   2     4-68  548     4-68  576 
7380  =2   3     4-68  548     4-68  576 
7440  =  2   4     4-68  548     4-68  576 
75oo  =25     |48  54^    4-68577 

17 


750—800 


N 

L  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P  P 

750 

75i 
752 

753 
754 
755 
756 

757 
758 
759 

760 

761 
762 
763 
764 
765 
766 

767 
768 
769 

770 

771 
772 

773 
774 
775 
776 

777 
778 

779 
780 

781 
782 
783 
784 

785 
786 

787 
788 
789 

790 

791 
792 
793 
794 
795 
796 

797 
798 
799 
800 

87506 

512 

5i8 

523 

529 

535 

54i 

547 

552 

558 

6 
i  0.6 

2   1.2 

3  1.8 
4  2.4 
5  3-0 
6  3.6 

7  4-2  ' 
8  4.8 
9  54 

5 

i   0.5 

564 
622 
679 

737 
795 
852 

910 

967 

88024 

570 
628 
685 

743 
800 

858 

9J5 
973 
030 

576 
633 
691 

749 
806 
864 
921 
978 
036 

58i 
639 
697 

754 
812 
869, 
927 
984 
041 

587 
645 
703 
760 

818 
875 

933 
990 
047 

593 
651 
708 

766 
823 
881 

938 
996 
053 

599 
656 
714 

772 
829 
887 

944 
#°o1 

058 

604 
662 
720 

111 
835 
892 

950 

*oo7 
064 

610 
668 
726 

783 
841 
898 

955 
*oi3 
070 

616 
674 

73i 

789 
846 
904 
961 
*oi8 
076 

081 

087 

093 

098 

104 

no 

116 

121 

127 

133 

138 
195 

252 

309 
366 

423 
480 
536 
593 

144 

201 

258 

315 
372 
429 

485 
542 
598 

150 
207 
264 

321 

377 
434 

49  1 
547 
604 

156 
213 

270 

326 

383 
440 

497 
553 
610 

161 
218 

275 
332 
389 
446 
502 
559 
6i5 

167 
224 

281 

338 
395 
45i 
508 

564 
621 

173 
230 

287 

343 
400 

457 

513 
570 
627 

I78 

235 
292 

349 
406 

463 

519 
576 
632 

184 
241 

298 

355 
412 
468 

525 
58i 
638 

190 
247 
304 
360 
417 
474 
530 
587 
643 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

705 
762 

818 

874 
930 
986 

89  042 
098 
154 

711 

767 
824 

880 
936 
992 

048 
IO4 
159 

717 

773 
829 

885 
941 
997 

053 
109 

165 

722 
779 
835 
891 

947 
*003 

059 
H5 
170 

226 

728 

784 
840 

897 
953 
*oog 
064 

120 
I76 

734 
790 
846 
902 
958 

3frOI4 

070 
126 

182 

739 
795 
852 

908 
964 

*020 
076 
131 
I87 

745 
801 

857 

913 
969 

*025 

081 
137 
193 

750 
807 
863 

919 

975 
4*031 
087 

143 
198 

756 
812 
868 

925 
981 
*037 
092 
148 
204 

209 

215 

221 

232 

237 

243 

248 

254 

260 

265 
321 
376 

432 
487 
542 

597 
653 
708 

271 

326 

382 

437 
492 

548 
603 
658 
713 

276 
332 

387 

443 
498 
553 
609 
664 
719 

2«2 

337 
393 
448 
504 
559 
614 
669 
724 

287 

343 
39« 
454 
509 
564 
620 
675 
730 

293 
348 
404 

459 
515 
570 
625 
680 
735 

298 

354 
409 

465 
520 

575 

631 
686 
741 

304 
360 

415 
470 
526 
58i 
636 
69r 
746 

310 
365 
421 

476 
53i 
586 

642 
697 

752 

315 
371 
426 

481 
537 
592 

647 
702 

757 

2    1.0 

3   1-5 
4   2.0 
5   2.5 
6   3.0 

•  7   3-5 
8   4.0 

9   4-5 

763 

768 

774 

779 

78! 

790 

796 

801 

807 

812 

8x8 

873 
927 

982 
90037 
091 
146 

200 

255 

•823 
878 
933 
988 
042 
097 

I5i 
206 
260 

829 
883 
938 

993 
048 

IO2 

157 
211 
266 

834 
889 

944 
998 

053 
108 

162 
217 
271 

840 
894 
949 
*oo4 
059 
H3 
168 

222 
276 

845 
900 

955 
3,009 
064 
119 

173 
227 
282 

85* 
9°5 
960 

*oi! 
069 
124 
179 

233 

287 

856 
911 
966 

*020 

075 
129 

184 
238 
293 

862 
916 
971 
#026 
'080 
135 
189 

244 

298 

352 

867 
922 
977 

*03i 
086 
140 

!95 
249 
304 

309   3H 

325 

331 

336 

342 

347 

358 

N 

L  0 

1 

234 

5 

7  |  8 

9 

P  P 

7500"  =2°  5'  S  4.68548  T  4.68577 
7560  =26     4.68  548     4.68  577 
7620  —  2  7     4.68  548     4.68  577 
7680  =28     4.68  547     4.68  578 
7740  =29     4.68  547     4.68  578 

7800"  =  2°  10'  S  4-68547  T  4-68578 
7860  =  2  ii     4.68  547     4.68  579 
7920  =  2  12     4.68  547  •   4.68  579 
7980  =  2  13     4-68  547     4.68  579 
8040  =  2  14     4.68  546     4.68  579 

18 


800—850 


N 

L  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P  P 

800 

801 
802 
803 
804 
805 
806 
807 
808 
809 

810 

Bn 

812 
813 
814 

815 
816 

817 
818 
819 

820 

821 
822 
823 
824 
825 
826 

827 
828 
829 

830 

831 
832 
833 
834 
835 
836 

837 
838 

839 
840 

841 
842 
843 
844 
845 
846 

847 
848 

849 

h.'O 

90309 

3H 

320 

325 

33i 

336 

342 

347 

352 

358 

6 

i   0.6 

2    1.2 

3   1.8 
4   2.4 
5   3-0 
6   3.6 

7   4-2 
8   4.8 
9   5-4 

5 

i   0.5 

2    1.0 

3   1.5 
4   2.0 
5  "2.5 
6   3.0 

7   3-5 
8   4.0 

9   4-5 

• 

363 
417 
472 
526 
580 
634 
687 
741 

795 

369 
423 
477 
531 
58! 
639 
693 
747 
800 

374 
428 
482 

536 
590 
644 
698 
752 
806 

380 

434 
488 

542 
596 
650 

703 
757 
811 

38! 
439 
493 

547 
601 

655 
709 
763 
816 

39° 

445 
499 

553 
607 
660 

714 

768 
822 

396 
450 
504 
558 
612 
666 
720 
773 
827 

401 

455 
509 
563 
617 
671 

725 
779 
832 

407 
461 
5i5 
569 
623 
677 
730 
784 
838 

412 
466 
520 

574 
628 
682 
736 
789 
843 

849 

854 

859 

865 

870 

875 

SSi 

886 

891 

897 

902 
956 
91009 

062 
116 
169 

222 

275 

328 

907 
961 
014 
068 

121 
174 
228 
28l 

334 

9*3 
966 
020 

073 
126 
1  80 

23-3 
286 

339 

gi& 
972 
025 

078 
132 
185 
238 
291 
344 

924 

977 
030 

084 

137 
190 

243 
297 
350 

929 
982 
036 
089 
142 
196 

249 
302 

355 

934 
988 
041 

094 
148 

201 

254 
307 
36O 

940 
993 
046 

IOO 

153 
206 

259 
312 
365 

945- 
998 
052 

105 
158 

212 
26§ 
318 
371 

950 
*oo4 
057 
no 

164 

217 
270 
323 

376 

38l 

387 

392 

397 

403 

408 

413 

418 

424 

429 

"^34 
487 
540 

593 
645 
698 

751 
803 
855 

440 
492 
545 
598 
651 
703 
756 
808 
861 

445 
498 

551 
603 
656 
709 
761 
814 
866 

450 
503 
556 
609 
661 
714 
766 
819 
871 

455 
508 
56i 
614 
666 
719 

772 
824 
876 

461 

5H 

566 

619 
672 
724 

777 
829 
882 

466 

519 
572 
624 
677 
730 
782 

834 
887 

47i 
524 

577 
630 
682 
735 

787 
840 
892 

477 
529 
582 

635 
687 
740 

793 

845 
897 

482 
535 
587 
640 
693 
745 

798 
850 
903 

908 

913 

918 

924 

929 

934 

939 

944 

9§o 

955 

960 
92012 
,   065 
117 
169 

221 

273 
324 
376 

965 
018 
070 

122 

174 
226 

278 
330 
381 

971 
023 
075 
127 
179 
231 
283 
335 
387 

976 
028 
080 

132 
184 
236 
288 
340 
392 

981 

033 

085 

137 
189 
241 

293 
345 
397 

449 

986 
038 
091 

143 
195 
247 
298 
350 
402 

991 
044 
096 
148 
200 
252 

304 

355 
407 

997 
049 

101 

153 
205 

257 

309 
361 
412 

*002 

054 
I  O6 

158 
2IO 
262 

314 
366 
418 

*°07 
059 
in 

163 
215 
267 

319 
37i 
423 

428 

433 

438 

443 

454 

459 

464 

469 

474 

480 
531 
583 
634 

686 
737 
788 
840 
891 

485 
536 
588 

639 
691 

742 

793 
845 
896 

49° 

542 
593 

645 
696 

747 

799 
850 
901 

495 
547 
598 
650 
701 
752 
804 
855 
906 

500 
552 
603 

655 
706 
758 
809 
860 
911 

505 
557 
609 

660 
7ii 
7^>3 
814 
865 
916 

5H 
562 
614 

665 
716 
768 
819 
870 
921 

5i6 
567 
619 

670 
722 

773 
824 

875 
927 

521 
572 
624 

675 
727 
778 
829 

881 
932 

526 
578 
629 

68  1 
732 
783 

834 
886 

937 

M- 

947 

952 

957 

962 

967 

973 

978 

983 

988 

N 

L  0 

1 

5 

3 

4 

5 

6 

7 

8 

o 

1'  P 

7980*  =2°  13'  S  4.68547  T  4.68579   8280*  =2°  18'  S  4.68546  T  4.68581 

8040  =2  14      4.68546      4.68579    8340  =2  19      4.68546      4.68581 
8100  =2  15      4.68546      4.68580    8400  =2  20      4.68545      4.68582 
8160  =2  16      4.68546      4.68580    8460  =2  21      4-68545      4.68582 
8220  =2   17      4.68546      4.68580    8520  =2  22      4-68545      4.68582 

19 


850—900 


N 

L  0 

1 

o 

3 

4 

5 

6 

7 

8  |  9 

P  P 

850 

851 
852 

853 
854 
855 
856 

857 
858 

859 
860 

861 
862 
863 
864 
865 
866 
867 
868 
869 

870 

871 
872 
873 
874 
875 
876 

877 
878 
879 

880 

881 
882 
883 
884 
885 
886 
887 
888 
889 

890 

891 
892 
893 
894 

895 
896 

897 
898 
899 

900 

92942 

947 

952 

957 

962 

967 

973 

978 

9S3 

988 

I 

2 

3 

4 
5 
6 

8 
9 

I 

2 

3 
4 

6 

7 
8 

9 
I 

2 

3 
4 
5 
6 

8 
9 

6 

0.6 

1.2 

1.8 
2.4 
3-0 

3-6 
4.2 
4.8 
54 

5' 

0.5 

I.O 

i-5 

2.0 

2-5 

3.0 

3-5 
4.0 

4-5 

4 
0.4 
0.8 

1.2 

1.6 

2.0 
2.4 

2.8 

3-2 
3.6 

993 
93044 
09! 
146 
197 
247 
298 
349 
399 

998 
049 

IOO 

151 

2O2 
252 

303 

354 
404 

*oo3 
054 
105 

156 
207 
258 
308 

359 
409 

*ooS 

059 
no 

161 

212 

263 

313 
364 
4U 

*oi3 
064 

H5 
1  66 
217 
268 

3i8 
369 
420 

*oi8 
069 

120 

171 
222 

273 
323 

374 
425 

*024 

075 
125 
176 
227 
278 
328 

379 
430 

*029 
080 

131 
181 
232 
283 

334 
384 
435 

*Q34 
085 
136 
1  86 

237 
288 

339 
389 
440 

*039 
090 
141 
192 
242 
293 
344 
394 
445 

450 

455 

460 

46! 

470 

475 

480 

485 

490 

495 

500 

551 
601 

651 

702 

752 
802 
852 
902 

505 
556 
606 

656 
707 
7.57 
807 

857 
907 

5io 
56i 
611 

66  1 
712 
762 
812 
862 
912 

515 
566 
616 
666 
717 
767 
817 
867 
917 

520 

57i 
621 

671 
722 
772 
822 
872 
922 

526 
576 
626 

676 
727 
777 
827 
877 
927 

53i 
58i 
631 
682 
732 
782 

832 
882 
932 

536 
586 
636 

687 

737 
787 

837 
887 

937 

54i 
59i 
641 

692 
742 

792 

842 
892 
942 

546 
596 
646 

697 
747 
797 
847 
897 
947 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

94  002 
052 

101 

151 
201 
250 
300 
349 
399 

007 

057 
1  06 

156 
206 

255 
305 
354 
404 

OI2 
O62 
III 

161 

211 
26O 
310 

359 
409 

017 
067 
116 
1  66 
216 
265 

315 
364 
4U 

022 
072 
121 

171 
221 
270 
320 

369 
419 

027 

077 
126 

176 
226 

275 

32| 
374 
424 

032 
082 
131 
181 
231 
280 

330 
379 
429 

037 
086 
136 
1  86 
236 
285 

335 
384 
433 

042 
091 
Mi 
191 
240 
290 
340 
389 
43S 

047 
096 
146 
196 

245 
295 
345 
394 
443 

448 

453 

458 

463 

46S 

473 

478 

483 

488 

493 

498 
547 
596 

645 
694 

743 
792 
841 
890 

503 

552 
601 

650 
699 
748 

797 
846 
895 

507 
557 
606 

655 
704 
753 
802 

851 
900 

512 
562 
611 
660 
709 
758 
807 
856 
9°5 

517 
567 

6t6 

665 
7M 
763 
812 
861 
910 

522 

5/i 
621 

670 
719 

768 

817 
866 
915 

527 
576 
626 

675 
724 
773 
822 

871 
919 

532 
58i 
630 

680 
729 
778 
827 
876 
924 

537 
586 

635 
685 
734 
783 
832 
880 
929 

542 

591 
640 

689 
738 
787 
836 
885 
934 

939 

944 

94.9 

954 

959 

963 

968 

973 

978 

983 

988 
95036 
085 

134 
182 
231 

279 

328 
376 

993 
041 
090 

139 
187 
236 
284 
332 
38i 

998 
046 
09! 

M3 
192 
240 
289 

337 

386 

*002 

051 

IOO 

148 

*97 

245 

294 

342 
390 

*oo7 
056 
105 

153 
202 
250 

299 
347 

395 

*OI2 

061 

109 

158 

207 

255 
303 
352 
400 

*oi7 
066 
114 

163 

211 
260 
308 

357 
405 

*022 

071 
119 

168 
216 

265 

313 
361 
410 

*027 

0/5 
124 

173 

221 
270 

318 
366 
415 

#032 
080 
129 

177 
226 
274 

323 
37i 
419 

424 

429 

434 

439 

444 

448 

453 

45S 

463 

468 

N 

L  0 

1  |  2 

3 

4 

5 

6 

7 

8    9 

P  P 

8460'  =2°  21'     84.68545   T4.6S532      8760"  =2°  26'    84.68544  T  4.68584 

8520  =2  22      4.68  545    4.68  582     8820  =  2  27     4.68  544    4.68  584 
8580  =  2  23      4.68  545    4.68  583     8880  =  2  28     4.68  544    4.68  584 
8640  -  2  24      4.68  545    4.68  583     8940  =  2  29     4.68  544    4.68  585 
8700  =  2  25      4.68  545    4.68  583     9000  =  2  30     4.68  544    4.68  585 

900—950 


M 

L  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

r  r 

900 

901 

902 
903 
904 

9°5 
906 

907 
908, 
909 

910 

911 
912 
9i3 
914 

9*5 
916 

917 

918 
919 

920 

921 
922 
923 
924 

925 
926 

927 
928 
929 
930 

93i 
932 
933 
934 
935 
936 

937 
938 
939 

940 

95424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

472 
521 
569 
617 
665 
713 
761, 
809 
856 

477 
525 
574 
622 
670 
718 
766 

8i3 
861 

482 

530 
578 
626 
674 
722 

770 

818 
866 

487 
535 
583 
631 
679 
727 

775 
823 
871 

492 
540 

588 

636 
684 

732 
780 
828 
875 

497 
545 
593 
641 
689 
737 

78! 
832 
880 

501 
550 
598 
646 

694 
742 

789 
837 

885 

506 

554 
602 

650 
698 

746 

794 
842 
•$9~6 

5ii 
559 
607 

655 
703 
75i 

799 
847 

895 

516 
564 
612 

660 
708 
756 
804 

S52 
899 

5 

i  0.5 

2   I.O 

3  1-5 

4   2.0 

5  2.5 
6  3.0 

7  3-5 
8  4.0 

9  4-5 

4 
i  0.4 

2   0.8 

3  1.2 
4  1.6 
5  2.0 
6  2.4 
7  2.8 
8  3-2 
9  3-6 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

952 

999 
96047 

095 
142 
190 

237 
284 
332 

957 
*oo4 
052 
099 

M7 
194 

242 
289 
336 

961 
*oo9 
057 
104 
152 
199 

246 
294 
34i 

966 
*oi4 
06  1 

109 
156 

204 

251 
298 
346 

971 
*°I9 
066 

114 

161 
209 
256 
303 
350 

976 
3,023 
071 

sx'8 

1  66 
213 
261 
308 
355 

980 

*°28 

076 
123 

171 
218 

265 

313 
360 

985 
*033 
080 

128 

175 
223 

270 

317 
36! 

990 
*038 
085 

133 
1  80 

227 

275 
322 

369 

995 

*042 

090 

137 

185 
232 
280 
327 
374 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

426 

473 
520 

567 
614 
661 

708 

755 
802 

43i 

4/8 
525 
572 
619 
666 

713 
759 
806 

435 
483 
530 

577 
624 
670 
717 
764 
811 

440 
487 
534 
58i 
628 
675 
722 
769 
816 

445 
492 

539 
586 

633 

680 

727 

774 
820 

450 
497 
544 

59i 
638 
685 

73i 

778 
825 

454 
501 
548 

595 
642 
689 

736 
783 
830 

459' 
506 

553 
600 
647 
694 

74i 

788 

834 

464 

5H 
558 
605 
652 
699 

745 
792 
839 

468 

515 
562 

609 
656 
703 
750 
797 
844 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

895 
942 

988 

97035 
08  1 

i2a 

174 
220 
267 

900 
946 
993 

039 
086 

132 

179 
225 
271 

904 
95i 
997 
044 
090 
137 
183 
230 
276 

909 
956 

*002 

049 
095 
142 

188 

234 
280 

914 
960 
*°°7 
053 

100 

146 
192 

239 

285 

918 
965 
*011 
058 
104 
151 
197 
243 
290 

923 
970 
*°l6 
063 
109 
155 
202 
248 
294 

928 
974 

*°2I 

067 
114 
1  60 
206 

253 
299 

932 
979 

*025 

072 
118 
165 

211 

257 
304 

937 
984 
*030 

077 
123 
169 
216 
262 
308 

313 

317 

322 

327 

33i 

336 

340 

345 

350 

354 

941 
942 
943 
944 
945 
946 

947 
948 
949 

359 
405 
451 
497 
543 
589 

635 
68  1 
727 

364 
410 
456 
502 
548 
594 
640 
685 
731 

368 
4U 
460 

506 

552 
598 

644 
690 
736 

373 
419 

465 
5ii 
557 
603 

649 
695 
740 

377 
424 
470 

5i6 
562 
607 

653 
699 

382 
428 
474 
520 
566 
612 
658 
704 
_749_ 
795 

387 
433 
479 

525 
57i 
617 

663 
708 

7?4 

391 
437 
483 
529 
575 
621 

667 
713 
759 

396 
442 

488 

534 
580 
626 
672 
7i- 
.763. 
809 

400 

447 
493 

539 
585 
630 

676 
739 

768 

950 

772 

111 

782 

:>''• 

791 

Boo 

804 

8i3 

N 

L  0 

1    t 

i 

4 

r,   <;   7   8 

D 

I'  P 

9000*  —2°  30'  S  4.68544  T  4.68585 

9060  =2  31      4-68544      4.68585 
9120  =  2  32      4.68543      4.68586 

9180  =2  33    4.68543     4.68586 
9240  =2  34    4.^8543     4-68587 

9300*  =2°  35'  84.68543  T  4.68587 

9360  =2  36      4.68543       4.68587 

to  =2  37    4.68542     4.68588 

9480  =2  38      4.68542      4.68588 

9540  =2  39    4.68  542     4.68  588 

21 


950—1000 


N 

L  0 

1  |  2 

3 

4 

5 

6 

7 

8 

Q 

P  P 

950 

95i 
952 
953 
954 
955 
956 

957 
958 
959 

960 

961 

962 
963 
964 

965 

966 

967 
968 
969 

970 

97i 
972 

973 
974 
975 
976 

977 
978 

979 
980 

981 
982 

983 
984 

985 
986 

987 
988 
989 

990 

991 
992 
993 
994 
995 
996 

997 
998 
999 
1000 

97772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

5 

i  0.5 

2   I.O 

3   i-5 
4  2.0 
5  2.5 
6  3.0 

7  3-5 
8  4.0 

9  4-5. 

4 

i  0.4 

2   0.8, 

3  1.2 
4  i-6 
5  2.0 
6  2.4 

7  2.8 
8  3.2 
9  3-6 

818 
864 
909 

955- 
98000 
046 
091 

137 

182 

823 
868 
914 

959 
005 
050 

096 
141 

1  86 

827 

873 
918 

964 
009 

055 

IOO 

146 
191 

832 
877 
923 
968 
014 
059 
105 
150 
i95 

836 
882 
928 

973 
019 
064 

109 

155 

200 

841 
886 
932 

978 
023 
068 

H4 
159 
204 

845 
891 

937 
982 
028 
073 
118 
164 
209 

850 
896 
941 
987 
032 

078. 

123 

168 
214 

855 
900 
946 
991 

037 

082 

127 

i73 

218 

859 
905 
950 
996 
041 
087 
132 
i77 
223 

227 

232 

236 

241 

245 

250 

254 

259 

263, 

268 

272 
3i8 
363 
408 

453 
498 

543 

588 
632 

277 
322 
367 
412 

457 
502 

547 
592 
637 

281 
327 
372 

4i7 
462 

507 
552 
597 
641 

286 

33i 
376 
421 
466 
5ii 
556 
60  1 
646 

290 
336 
381 
426 
471 

51,6 

56i 
605 
650 

29! 
340 
385 
430 
475 
520 

565 
610 

655 

299 
345 
39° 

435 
480 

525 
570 
614 

659 

304 
349 
394 

439 
484 

529 

574 
619 
664 

308 
354 
399 

444 

489 

534 

579 
623 
668 

313 

358 
403 
448 
493 
538 

583 
628 

673 

6/7 

682 

686 

691 

695 
740 

784 

829 

874 
918 

963 
*oo7 
052 
096 

700 

704 

709 

7J3 

7i/ 

722 

767 
811 

856 
900 
945 
989 

99034 
078 

726 

77i 
816 

860 
90! 
949 

994 

038 
083 

73i 

776 
820 

865 
909 
954 
998 

043 

087 

735 
780 
825 
869 
914 
958 

*o°3 
'047 
092 

744 
789 

834 

878 

923 
967 

*OI2 
056 
IOO 

749 
793 

838 

083 
927 
972 
.X.oi6 
"06  1 
105 

753 
798 

843 
887 
932 
976 

*°2I 
065 
I09 

758 
802 
847 
892 
936 
981 

*025 

069 

114 

762 
807 
85  v 
896 
94i 
985 

<-029 
074 

118 

123 

127 

131 

136 

140 

145 

M9 

154 

158 

162 

167 

211 

255 
300 

344 

388 

432 
476 
520 

171 
216 
260 

304 
348 
392 
436 
480 
524 

176 
220 
264 
308 
352 
396 
441 

484 

528 

1  80 
224 
269 

313 
357 
401 

445 
489 

533 

185 
229 

273 

317 
361 
405 

449 

493 
537 

189 
233 
277 
322 
366 
410 

454 
498 

542 

193 

238 
282 

326 
3/0 
414 

458, 
502 
546 

I98 
242 

286 

330 

374 
419, 
463 
506 

550 

202 

247 
291 

335 
379 
423 

467 
5ii 

555 

207 
251 
295 

339 
383 
427 
47i 

515 
559 

564 

568 

572 

577 

58i 

585 

590 

594 

599 

603 

607 
651 
69! 

739 
782 
826 
870 
913 
957 

612 
656 
699 

743 
787 
830 

874 
917 
961 

616 
660 
704 

747 
79  1 
835 
878 
922 
965 

621 
664 
708 
752 
795 
839 
883 
926 
970 

6?5 
669 
712 

756 
800 
843 
887 
930 
974 

629 

673 
717 

760 
804 
848 

891 

935 

978 

634 
677 
721 

765 
808 
852 
896 
939 
983 

638 
682 
726 

769 
813 
856 

900 
944 
987 

642 

686 
730 

774 
8-17 
861 

904 
948 
991 

647 
%i 
734 
778 
822 
865 
909 
952 
996 

ooooo 

004 

009 

013 

017 

022 

026 

030 

035 

039 

5f 

L  0    1  |  2 

3    4 

567 

8 

9 

P  P, 

9480"  —2°  38'  S  4.68  542  T  4.68  588 
9540  =2  39    4.68542     4.68588 

9600  =2  40      4.68542      4.68589 
9660  =2  41      4.68542      4.68589 
972O  =2  42      4.68541      4.68590 

9780"—  2°  43'  S  4.68541  T  4.68590 
9840  =2  44    4.68541     4.68590 
9900  —2  45     4-68541     4-68591 
9960  =2  46    4.68541     4.68591 
10020  =2  47     4.68540     4.68592 

22 


THE   NATURAL    LOGARITHMS 

OF 

WHOLE  NUMBERS  FROM  1  TO  200. 

Common  logarithms  may  be  converted  into  natural  logarithms  by  multi- 
plying them  by  2.3025850930. 

Natural   logarithms   may   be   converted   into    common    logarithms   by 
multiplying  them  by  0.4342944819. 

N 

Nat  Log 

N 

Nat  Log 

N      Nat  Log 

N 

Nat  Log 

:NT 

Nat  Log 

0 

i 

2 

3 
4 

6 

8 
9 
10 
ii 

12 
13 
14 
15 

16 

17 
18 

19 
20 

21 
22 
23 
24 
25 
26 

2? 
23 

29 

30 

3i 
32 

33 
34 
35 
36 

37 
38 
39 

40 

—  oo 

40 

4i 
42 

43 
44 
45 
46 

47 
48 

49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 

60 

61 
62 
63 

64 
65 
66 

67 
68 
69 

70 

7i 
72 
73 
74 
75 
76 

77 
78 
79 
80 

3.68  888 

80 

81 
82 
83 
84 

85 
86 

87 
88 
89 

90 

91 
92 

93 
94 
95 
96 

97 
98 

99 
100 

101 
102 
103 
104 
105 

106 
107 
108 
109 

110 
in 

112 
113 

I  I-l 
U5 

1x6 

"7 
118 
tig 

]'2n 

4-38  203 

120 

121 
122 
123 
124 

125 
126 

127 
I2S 
I29 

130 

131 
132 
133 
i34 
i35 
136 

i37 
138 

139 
140 

141 
142 

143 
144 

145 
146 

147 
148 
149 

150 

151 
152 
153 
154 
155 
156 

J57 
158 
'59 

1GO 

4.78  749 

160 

161 

162 
163 
164 
165 
1  66 

167 
1  68 
169 

170 

171 
172 

173 
174 
175 
176 

177 
178 
179 

180 

181 
182 
183 
184 

185 
186 

187 
188 
189 

190 

5-07517 

O.OOOOO 
0.69315 
1.09  86  1 
1.38  629 
1.60944 
1.79176 
1.94  591 
2.07  944 
2.19  722 

3-71  357 
3-73  767 

3.76  I2O 
3.78419 

3.80  666 
3.82  864 

3-85015 

3.87  I2O 
3.89182 

4-39  445 
4.40  672 
4.41  884 

4-43  082 
4.44265 
4-45  435 
446  59i 
4-47  734 
4.48  864 

4-79  579 
4.80  402 
4,81  218 
4.82  028 
4.82  831 
4.83  628 
4.84  419 
4.85  203 
4.85  981 

5.08  140 
5.08  760 
5-09375 

5-09  987 
5-10595 
5.11  199 

5-  1  1  799 
5-12  396 
5-12990 

2.30  259 

3.91  202 

4.49981 

4-86  753 

5-i358o 

2.39  790 
2.48  491 
2.56  495 
2.63  906 
2.70  805 
2.77  259 
2.83  321 
2.89037 
2.94  444 

3-93  183 
3-95  124 
3-97  029 
3.98  898 
4-oo  733 
4-02  535 

4-04  305 
4.06  044 
4-07  754 

4.51  086 
4.52  179 
4-53  260 
4-54  329 
4-55  388 
4-5643! 
4.57471 
4.58497 
4-59512 

4-87520 
4.88  280 
4-89035 

4-89  784 
4.90  527 
4.91  265 
4.91  998 
4-92  725 
4-93  447 

5.14  166 
5.14  749 
5.15329 

5-15  906 
5.16479 
5-17048 

5-17615 

5.18  178 

5.18739 

2-99  573 

4-09  434 

4-60517 

4-94  164 

5-19296 

3-04  452 
3-09  104 
3.13  549 
3-17805 
3.21  888 
3.25  810 

3-29  584 
3-33  220 
3.36  730 

4.11  087 
4-12713 
4.I43I3 
4.15  888 

4-17439 
4.18965 

4.20  469 
4.21951 
4.23411 

4.61  512 
4.62  497 
4.63  473 
4.64  439 

4.65  396 
4.66  344 

4.67  283 
4.68  213 
4-69  135 

4-94  876 
4-95  583 
4.96  284 

4.96  981 
4-97  673 
4.98  361 

4-99  043 
4-99  721 
5-00395 

5-19850 
5.20401 
5-20  949 

5-21  494 
5.22036 
5-22  575 

5-23111 
5.23  644 
5.24175 

340  120 

4.24  850 

4.70048 

5.01  064 

5-24  702 

343  399 
3.46574 
3.49651 

3.52  636 
3-55  535 
3.58352 
3.61  092 
3.63  759 
3.66356 

4.26  268 
4.27  667 
4.29  046 

4.30407 
4Oi  749 
4-33  073 
4.34381 
4.35671 
4O6945 

4.70953 
4-71  850 
4-72  739 
4-73  620 
4-74493 
4-75  359 
4.76217 
4.77068 
4.77912 

5.01  728 
5.02  388 
5-03  044 
5-03  695 
5.04  343 
5.04  986 

5.05  62§ 
5.06  260 
5.06890 

191 
192 
i93 
194 

195 
196 

197 
198 
199 

200 

5-25  227 
5-25  750 
5.26  269 

5.26  786 
5-27  300 
5.27811 
5.28  320 
5.28827 
5-29  330 

3.68  888 

4-38  203 

4.78  749 

5.07517 

5.29  832 

II 

TABLE  OF  ADDITION  AND  SUBTRACTION 
LOGARITHMS 

FOR  THE 

CALCULATION  OF  THE  LOGARITHMS 

OF  THE 

SUM  AND  DIFFERENCE  OF  TWO  NUMBERS  WHOSE 
LOGARITHMS  ARE  GIVEN. 


24 


ADDITION. 

A 

B  0 

1 

2 

3 

4 

5 

6 

7 

8 

y 

P  P 

0.00 

.30  103 

053 

003 

J.-953  1*903 

*S54 

*8o4 

*754 

*705 

*655 

OI 

.29  606 

556 

507 

458 

409 

359 

310 

261 

212 

163 

50 

49 

48 

47 

02 

115 

066 

017 

*968 

*92° 

#871 

$.822 

*774 

#726 

3.677 

i 

5-0 

4.9 

4.8 

4-7 

03 

0.28  629 

581 

532 

484 

436 

388 

340 

292 

245 

197 

2 

IO.O 

9.8 

9.6 

9-4 

04 

149 

101 

054 

006 

*959 

*9IZ 

#864 

*8i7 

^769 

*722 

3 

15.0 

14.7 

14.4 

14.1 

05 

0.27675 

628 

534 

487 

440 

393 

346 

300 

253 

4 

2O.O 

19.6 

19.2 

18.8 

06 

207 

1  60 

114 

067 

021 

*974 

*928 

*79° 

5 

25.O 

24-5 

-24.0 

23-5 

07 
08 
09 

0.26  744 
287 
0.25  836 

698 
242 
791 

652 
196 
746 

606 

151 
701 

560 
1  06 
657 

515 
06  1 
612 

469 
016 

568 

423 
*9/o 
523 

3/8 
479 

332 
*88i 
434 

6 

7 
3 
9 

30.0 
35-0 
4O.O 

29.4 
34-3 
39-2 
44.1 

28.8 
33-6 
38.4 
43.2 

28.2 
32.9 
37-6 
42.3 

0.10 

390 

346 

302 

258 

214 

170 

126 

082 

038 

*994 

ii 

0.24  950 

907 

863 

819 

7/6 

733 

689 

646 

603 

559 

46 

45 

44 

43 

12 

473 

430 

387 

344 

301 

258 

216 

173 

130 

i 

4.6 

4-5 

44 

4-3 

13 

088 

045 

003 

#960 

*9J8 

*833 

*79r 

*749 

2 

9-2 

9.0 

8.8 

8.6 

14 
15 

0.23665 
247 

623 
206 

58i 
165 

539 
123 

497 
082 

455 
041 

414 

000 

372 
*959 

330 

*9l8 

289 

*877 

3 
4 

13.8 
18.4 

13.5 
1  8.0 

13-2 
17.6 

12.9 

I-.2 

16 

17 

18 

0.22  836 

430 

O2Q 

795 

389 

754 
349 

713 
309 

673 
269 

632 
229 

591 
189 

149 
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109 

470 
069 

5 

6 

7 

23.0 

27.6 
32.2 

„£  0 

22.5 
27.0 

31-5 
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22.O 
26.4 
30.8 

21.5 
25-8 
30.1 

19 

0.21  634 

595 

556 

5i6 

477 

438 

399 

361 

322 

'283 

9 

41.4 

40.5 

35-2 

39-6 

34-4 
38.7 

0.20 

244 

206 

167 

128 

090 

052 

013 

*975 

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*898 

21 

O.2O86O 

822 

784 

746 

708 

670 

632 

594 

5S7 

519 

42 

41 

40 

30 

22 

481 

444 

406 

331 

294 

220 

182 

145 

i 

4.2 

4.1 

4.0 

3-9 

23 

108 

071 

034 

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#960 

*923 

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2 

8.4 

8.2 

8.0 

7.8 

24 

25 
26 

0.19  740 

378 

020 

704 
342 

667 
306 
*949 

631 
270 

595 
234 
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558 
198 
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522 
163 

486 
127 
*773 

450 
091 

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414 
056 
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j 

4 

f> 

12.6 

16.8 

2I.O 
25.2 

12.3 
16.4 
20.5 
24.6 

12.0 

16.0 

2O.O 
24.O 

11.7 
15-6 
19-5 
23.4 

27 
28 

0.18  668 
322 

633 
287 

599 
253 

564 
218 

529 
184 

494 
150 

460 
116 

425 
082 

390 
048 

356 
014 

7 
8 

29.4 
336 

28.7 
3?  8 

28.0 
32.O 

27-3 
31.2 

29 

0.17980 

946 

912 

878 

845 

Sii 

777 

744 

710 

677 

9 

37-8 

36.9 

36.0 

35-1 

0.30 

643 

610 

577 

544 

510 

477 

444 

411 

378 

345 

3i 

312 

279 

247 

214 

181 

148 

116 

083 

051 

018 

38 

37 

36 

35 

32 

0.16986 

954 

921 

889 

857 

825 

793 

761 

729 

697 

i 

3-» 

3-V 

3-6 

3-5 

33 

665 

633 

001 

569 

538 

506 

474 

443 

411 

380 

- 

7.6 

7-4 

TO  R 

7.0 

34 
35 
36 

349 
037 
0.15  73i 

317 
007 
701 

286 
*976 
670 

255 
*945 
640 

224 
*9X4 
610 

192 

*884 
580 

161 

*853 
550 

130 
520 

099 

%89 

068 
460 

3 

4 

6 

11.4 
15.2 
19.0 

22.8 

14.8 

18.5 

22.2 

14.4 

1  8.0 

21.6 

10.5 
14.0 
17.5 

2I.O 

37 

430 

400 

370 

340 

310 

281 

251 

221 

192 

162 

7 

26.6 

2S-9 

2S.2 

24.5 

38 

133 

104 

074 

045 

016 

^.986 

#957 

#928 

5*899 

.3.870 

S 

30.4 

29.6 

28.8 

28.0 

39 

0.14841 

812 

783 

755 

726 

697 

668 

640 

611 

583 

9 

•34.2 

33-3 

32.4 

31-5 

0.40 

554 

526 

497 

469 

441 

412 

384 

356 

328 

300 

4i 

272 

244 

216 

188 

160 

132 

104 

077 

049 

021 

34 

33 

32 

31 

42 

0.13994 

966 

939 

911 

884 

857 

829 

302 

775 

748 

i 

34 

3-3 

3-2 

3-1 

43 

721 

694 

667 

640 

613 

586 

559 

532 

505 

479 

2 

6.8 

6.6 

6.4 

6.2 

44 
45 
46 

47 
48 

452 
188 
0.12928 

673 
422 

425 
162 

903 
648 
397 

399 
136 
877 
622 
372 

372 
no 
851 

597 
348 

346 
084 
826 

572 
323 

319 

058 
800 

547 
298 

293 
032 

775 
522 
274 

267 
006 

749 
497 
249 

240 

724 
472 
224 

214 

»954 
698 

447 

200 

3 

4 
5 

7 
B 

10.2 
13.6 
17.0 
20.4 
23.8 
27.2 

9.9 
13-2 
16.5 
19.8 
23.1 
26.4 

9.6 

12.8 

16.0 
19.2 
22.4 
2S.6 

9.3 
12.4 

15.5 

1  8.6 

21.7 
24.8 

49 

175 

15* 

127 

[Ofl 

078 

"54 

030 

005 

#981 

#957 

9 

30.6 

20.7 

28.8 

27.0 

0.50 

0.11933 

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885 

861 

837 

814 

790 

766 

742 

719 

A 

B  0 

I 

2    3 

4 

5 

0 

7 

8    9 

P  P 

a>6,    A  =  log  a  —  log  6,    log  (a  -|-  6)  =  log  a-f-B. 

25 


ADDITION. 

A 

B  0 

1  |  2 

3 

4 

5« 

6  |  7 

8 

9 

P  P 

0.50 

o.i  i  933 

909 

885 

861 

837 

814 

79° 

766 

742 

719 

OA 

9Q 

9ft 

97 

5i 

69! 

671 

648 

624 

60  1 

577 

554 

531 

507 

484 

I 

0\J 

3.0 

LtV 
2.9 

£O 
2.8 

Ak 

2.7 

52 

461 

438 

415 

392 

368 

345 

323 

300 

277 

254 

2 

6.0 

5-8 

5.6 

54 

53 

»   231 

208 

186 

163 

140 

118 

095 

073 

050 

028 

3 

9.0 

8.7 

8.4 

8.1 

54 

005 

#983 

#960 

*938 

3,916 

^894 

*872 

*849 

*827 

45.805 

4 

12.0 

n.6 

II.  2 

10.8 

55 

o.io  783 

761 

739 

718 

696 

674 

652 

630 

609 

587 

5 

15.0 

M.5 

14.0 

13.5 

56 

565 

544 

522 

SGI 

479 

458 

437 

415 

394 

373 

6 

1  8.0 

17.4 

16.8 

1  6.2 

57 

330 

309 

288 

267 

246 

225 

204 

183 

162 

7 

2I.O 

20.3 

19.6 

18.9 

58 

141 

120 

100 

079 

058 

038 

017 

#996 

*976 

*955 

8 

24.0 

23.2 

22.4 

21.6 

59 

0.09935 

914 

894 

874 

853 

833 

813 

793 

773 

752 

9 

27.0 

26.1 

25.2 

24.3 

0.60 

732 

712 

692 

672 

652 

632 

612 

593 

573 

553 

t\f* 

Off 

c\  4 

OO 

61 
62 
63 

533 
338 
146 

5M 
319 

127 

494 
299 
108 

474 
280 
090 

455 
261 
071 

435 
242 
052 

416 
223 
033 

396 
204 
014 

377 
184 
*996 

357 
165 
*977 

i 

2 

T 

zo 

2.6 

5-2 

7-8 

25 

2.5 
5-0 

7.5 

24 

2.4 

4.8 

7-2 

23 

2.3 
4.6 
6.9 

64 

0.08  958 

940 

921 

902 

884 

865 

847 

829 

810 

792 

J 

4 

10.4 

IO.O 

9.6 

y 

9.2 

65 

774 

755 

737 

719 

701 

683 

664 

646 

628 

610 

5 

13.0 

12.5 

I2.O 

ii>5 

66 

592 

574 

557 

539 

521 

503 

485 

468 

450 

432 

6 

15-6 

15.0 

144 

13-8 

67 

415 

397 

379 

362 

344 

327 

309 

292 

275 

257 

7 

18.2 

17.5 

16.8 

16.1 

68 

240 

223 

206 

188 

171 

154 

137 

120 

103 

086 

8 

20.8 

2O.O 

19.2 

18.4 

69 

069 

052 

035 

018 

001 

*95i 

*934 

*9i8 

9 

23.4 

22.5 

21.6 

20.7 

0.70 

0.07  901 

884 

868 

851 

835 

818 

802 

785 

769 

753 

7i 

736 

720 

704 

687 

671 

655 

639 

623 

607 

591 

22 

21 

iy 

18 

T  R 

72 

575 

559 

543 

527 

495 

479 

463 

448 

432 

i 

2.2 

2.1 

1.9 

_  o 

1.0 
„  /• 

73 

416 

400 

38! 

369 

354 

338 

322 

307 

291 

276 

2 

44 

4.2 

/•   _ 

3-8 

3-6 

74 

261 

2^5 

230 

215 

199 

184 

169 

154 

138 

123 

3 
i 

6.6 

8.8 

6-3 

5.7 
7  6 

54 

7  2 

75 
76 

108 
0.06  959 

093 
944 

078 
929 

063 
914 

048 
900 

033 

885 

018 

870 

003 

856 

*84i 

*973 
827 

4 

5 
6 

II.O 

13.2 

10.5 
12.6 

/  *v 

9-5 
11.4 

/  • 
9.0 

10.8 

77 

812 

79s 

783 

769 

754 

740 

725 

711 

697 

683 

7 

154 

14.7 

13-3 

12.6 

78 

668 

654 

640 

626 

612 

597 

583 

569 

555 

54i 

8 

17.6 

16.8 

15.2 

14.4 

79 

527 

513 

500 

486 

472 

458 

444 

430 

417 

403 

9 

19.8 

18.9 

17.1 

16.2 

0.80 

389 

3/6 

362 

348 

335 

321 

308 

294 

281 

267 

81 

254 

240 

227 

214 

200 

187 

174 

161 

147 

134 

11 

16 

T  f\ 

15 

14 

I   A 

82 

121 

108 

09! 

082 

069 

056 

043 

030 

017 

004 

i 

*«/ 

1  .O 

I«5 

4 

n  Q 

83 

0.05  991 

978 

965 

952 

939 

927 

914 

901 

889 

876 

2 

T 

34 
e.I 

3-2 

4.8 

3-0 
4-5 

2.O 
4.2 

84 

863 

851 

838 

825 

8i3 

800 

788 

775 

763 

751 

O 

4 

*?• 

6.8 

6.4 

6.0 

5-6 

85 

738 

726 

714 

701 

689 

677 

664 

652 

640 

628 

5 

8-5 

8.0 

7.5 

7.0 

86 

616 

604 

591 

579 

567 

555 

543 

531 

519 

508 

6 

10.2 

9.6 

9.0 

8.4 

87 

•  496 

484 

472 

460 

448 

436 

425 

413 

401 

390 

7 

II.9 

II.  2 

10.5 

9.8 

88 

378 

366 

355 

343 

332 

320 

308 

297 

286 

274 

8 

13-6 

12.8 

12.0 

II.  2 

89 

263 

251 

240 

229 

217 

206 

195 

183 

172 

161 

9 

15-3 

14.4 

13-5 

12.6 

0.90 

i§o 

139 

127 

116 

105 

094 

083 

072 

061 

050 

91 

039 

028 

017 

006 

*995 

*985 

*974 

*963 

*952 

*94i 

13 

12 

11 

y 

92 

0.04931 

920 

909 

898 

888 

877 

867 

856 

845 

835 

i 

!>3 

1.2 

I.I 

0.9 

_   Q 

93 

824 

814 

.  803 

793 

782 

772 

762 

751 

731 

2 

2.6 

2.4 
a  6 

2.2 

3n 

1.0 

94 
95 

720 

618 

710 
608 

700 
598 

689 

588 

679 

578 

669 

568 

659 

558 

649 

548 

639 
538 

628 

528 

3 
4 

5 

3-9 

5-2 

6.5 

3-° 

4.8 
6.0 

O 

44 

5-5 

3^6 
4.5 

96 

519 

509 

499 

489 

479 

469 

460 

450 

440 

430 

6 

7.8 

7-2 

6.6 

54 

97 

421 

411 

401 

392 

382 

373 

363 

353 

344 

334 

7 

9.1 

8.4 

7-7 

6.3 

98 

325 

315 

306 

297 

287 

278 

268 

259 

250 

240 

8 

10.4 

9.6 

8.8 

7.2 

99 

231 

222 

213 

203 

194 

185 

176 

167 

157 

148 

9 

11.7 

10.8 

9.9 

8.1 

1.00 

139 

130 

121 

112 

103 

094 

085 

076 

067 

058 

A 

B  0 

1 

o 

3 

4 

5 

6 

7 

8 

9 

P  P 

a  >  6,    A=loga  —  log  &,    log  (a-j-6)  =  loga  +  B. 

ADDITION. 

A 

B  0 

1 

2 

3 

4 

5 

.6 

7 

8 

9 

P 

P 

1.00 

OI 
02 

03 
04 

05 
06 

07 
08 
09 

1.10 
ii 

12 
13 
14 
15 

16 

17 

18 

19 
1.20 

21 
22 
23 
24 

25 
26 

27 
28 

29 

1.30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 
1.40 

41 
42 
43 
44 
45 
46 

47 
48 
49 
1.50 

0.04  139 

130 

121 

112 

103 

094 

085 

076 

067 

058 

i 

2 

3 
4 
5 
6 

7 
8 

9 

8 

I   0. 
2   I. 

3  2. 
4  3- 
5  4- 
6  4. 

7  5. 
8  6. 

9  7- 

i 

2 

3 
4 
5 
6 

7 
8 

9 
5 

I   0. 
2   I. 

3   I- 
4  2. 
5  2. 
6  3- 
7  3- 
8  4 
9  4- 

i 

2 

3 
4 

6 

8 
9 

( 

0 

i 

2 

3 
4 

6 

8 

8 

6 

4 

2 
0 

S 
6 
4 

2 

( 

0 
I 
I 
2 
3 
3 
4 
4 
5 

5 
o 

5 

0 

5 

0 

5 

0 

5 
| 

0 

o 

0 

I 
I 
I 

2 

2 
2 

) 

•9 

.8 

•  7* 
.6 

•5 
•4 
•3 

.2 
.1 

7 
0.7 
1.4 

2.1 

2.8 

3.5 
4.2 

4.8 
5-6 
6.3 

) 
.6 

.2 

.8 
•4 
.0 
.6 

.2 

.8 
•4 

4 

0.4 
0.8 

1.2 

1.6 

2.O 
2.4 
2.8 

3.2 

3-6 

J 

•3 
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•9 

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!s 

.1 

4 
•  7 

049 
0.03  961 
875 
790 
708 
627 

548 
470 
394 

040 

953 
866 

782 
700 
619 

540 
462 

386 

032 

944 

858 

774 
691 

611 

532 
455 
379 

023 

935 
849 

765 
683 
603 

524 
447 
371 

014 
926 
841 

757 
675 
595 
516 
439 
364 

005 
918 
832 

749 

667 
587 
509 
432 

357 

*996 
909 
824 

741 
659 
579 
SGI 
424 
349 

*987 
901 
816 

732 
651 
571 
493 
417 
342 

*979 
892 
807 

724 
643 

563 

4S5 
409 

334 

*970 

883 

799 
7i6 
635 
555 
478 
401 
327 

320 

312 

305 

298 

290 

283 

276 

268 

261 

254 

247 

175 
1  06 

037 

0.02  971 
905 
84I 

779 
717 

240 
168 

099 

031 
964 
899 

835 
772 
7ii 

232 
161 
092 
024 

957 
892 

829 
766 

705 

225 
154 
085 
017 

95i 

886 

822 
760 
699 

218 

147 
078 

on 

944 

879 

816 
754 
693 

211 
140 
071 
004 

938 
873 

810 

748 
687 

204 

133 
065 

*997 
931 
867 

803 
742 
681 

197 
126 
058 

*99* 
925 
860 

797 
735 
675 

190 
1  20 
051 

*984 
918 

854 

79  1 
729 
669 

183 
H3 
044 

*977 
912 

848 

78! 
723 
663 

657 

651 

645 

639 

634 

628 

622 

616 

610 

604 

599 
54i 
485 
430 
376 
323 
272 

221 

172 

593 
535 
479 
424 
37i 
3i8 

267 
216 
167 

587 
530 
474 
419 
365 
313 
262 

211 
162 

58i 
524 
468 

414 
360 

308 

257 
207 
158 

575 
518 

463 

408 
355 
303 
252 

202 
153 

570 
513 

457 

403 
350 
297 
246 
197 
148 

564 
507 
452 

397 
344 
292 

241 
192 
143 

558 
502 
446 
392 
339 
287 

236 

187 
138 

552 
496 
441 

387 
334 
282 

231 
182 
133 

547 
490 

435 
33i 

329 
277 
226 

177 
129 

124 

119 

114 

no 

105 

TOO 

095 

091 

086 

081 

077 
030 

o.oi  985 

941 

898 
856 
814 

774 
734 

072 
026 
981 

937 

894 

851 
Sio 
770 
730 

067 
021 
976 

932 
889 
847 
806 
766 
726 

063 
017 
972 

928 
885 
843 
802 
762 
722 

058 

012 

967 
924 

881 
839 
798 
758 
719 

053 
008 

963 
919 

877 
835 

794 

754 
715 

049 
003 

959 

915 
872 
831 

790 
7§o 
7ii 

044 
*999 
954 
911 
868 
827 
786 
746 
707 

040 
*994 
950 
906 
864 
822 
782 
742 
703 

035 
*99° 
945 
902 
860 
818 

778 
738 
699 

695 

692 

688 

684 

680 

676 

673 

669 

665 

66  1 

658 
621 
5&4 

549 
5U 
480 

447 
415 
383 

654 
617 
581 

545 
5ii 
477 

444 
412 
380 

650 
613 
577 
542 
507 
474 
441 
408 
377 

646 
610 
574 
538 
504 
470 

437 
405 
374 

643 
606 
570 

535 
501 
467 

434 
402 

371 

639 
602 
566 

531 
497 
464 
431 
399 
368 

635 
599 
563 
528 

494 
460 

428 
396 
364 

632 
595 

559 

525 
490 

457 
424 
393 
361 

628 

591 
556 
521 
487 
454 
421 
389 
358 

624 

588 

552 

5i8 
484 
450 
418 
386 
355 

o.oi  352 

349 

346 

343 

340 

337 

334 

331 

328 

325 

A 

BO    1  |  2 

3 

4 

5 

6 

7 

8 

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P  P 

a>6,    A  —  log  a  —  log  b,    \og(a-+-b)  =  log  a-f-B. 

27 


ADDITION. 

A 

B  0 

1 

2 

3 

4 

5 

6 

7    8 

9 

P  P 

1.50 

52 
53 
54 
55 
56 

57 
58 
59 

1.60 

61 
62 
63 
64 
65 
66 

67 
68 
69 

1.70 

72 
73 
74 

75 
76 

77 
78 
79 
1.80 

81 

82 
83 
84 
85 
86 

87 
88 
89 

1.90 

91 
92 
93 
94 
95 
96 

97 
98 
99 

2.00 

o.oi  352 

349 

346 

343 

340 

337 

334 

33i 

328 

325 

3 

i  0.3 

2   0.6 

3  0.9 
4  1.2 

6  i!s 
7  2.1 
8  2.4 
9  2.7 

322 
292 

263 

235 

207 

1  80 

153 
128 

IO2 

319 
289 
260 
232 
204 
177 

125 

100 

316 

286 
257 
229 
202 
175 
148 

122 
097 

313 
283 

255 
226 
199 
172 
146 

120 

095 

310 
280 
252 
224 
196 
169 

143 
117 
092 

307 
278 

249 

221 
193 
167 
140 
115 
090 

304 
275 
246 

218 
191 
164 
138 

112 

087 

301 
272 
243 

215 
188 
161 

135 
no 
085 

298 
269 
240 

213 
185 
159 

133 
107 

082 

295 
266 

238 

2IO 

I83 
156 

130 

080 

o.oi  077 

075 

073 

O7O 

068 

065 

063 

060 

058 

056 

053 
030 
006 

o.oo  984 
962 

940 
919 

898 
878 

051 
027 
004 
981 

959 
938 

91? 

896 

876 

048 
025 
002 

979 
957 
936 

915 

894 

874 

046 
022 

*999 
977 
955 
933 
912 
892 
872 

044 
020 
*997 
975 
953 

910 

890 
870 

041 

018 
*995 
973 
951 
929 

908 

888 
868 

039 

016 
*993 
970 

948 
927 
906 
886 
866 

037 
013 

968 
946 
925 
904 

884 
864 

034 
on 

*988 
966 
944 
923 
902 
882 
862 

032 
009 

964 
942 
92I 

900 

880 
860 

o.oo  858 

856 

854 

852 

850 

848 

846 

844 

842 

84I 

839 
820 

801 

783 
766 
748 

731 
715 
699 

837 
818 

799 

764 
747 
730 
713 
697 

835 
816 
798 
780 
762 
745 
728 
712 
696 

833 
814 
796 

778 
760 
743 
726 
710 
694 

831 
812 

794 

776 
759 

725 
708 
692 

829 
810 
792 
774 
757 
740 

723 
707 
691 

827 
809 
790 

773 
755 
738 
721 

705 
689 

825 
807 
789 

753 
736 
720 

703 

688 

823 
805 
787 

769 

752 
735 

702 
686 

822 
803 
78! 
767 
750 

733 
716 
700 
684 

o.oo  683 

.681 

680 

678 

677 

675 

674 

672 

671 

669 

enenen  enoo  OOO 
enocoOOtOWenO 
OO  to  en  O  O3  oo  to  -^i 

666 

651 
636 

622 
608 
594 

58i 
567 

555 

664 
649 
635 
620 
606 
593 

579 
566 

553 

663 
648 
633 
619 
605 

578 
565 

552 

661 
646 
632 

618 
604 

59° 
577 
564 
551 

660 

645 
630 

616 
602 
589 

575 
562 
550 

658 
644 
629 

615 
601 

587 

574 
56i 

548 

657 
642 
628 

613 
599 
586 

573 
560 

547 

655 
641 
626 

612 

598 
585 

571 
558 
546 

654 
639 
625 

611 
597 

583 
570 
557 
545 

o.oo  543 

542 
530 

506 

495 
483 
473 
462 
451 
44i 

54i 

540 

538 

537 

536 

535 

533 

532 

531 
519 
507 
496 
485 

474 

463 
452 

442 

529 
517 
505 

494 

482 

471 
461 
450 
440 

527 
515 
504 
492 
481 
470 
460 
449 
439 

526 

503 

49  i 
480 
469 

459 

448 

438 

525 
513 
502 

490 

479 
468 

458 
447 

437 

524 

512 
500 

489 
478 
467 

457 
446 
436 

523 

499 

488 

477 
466 

456 
445 

435 

521 

498 
487 
476 
465 
454 
444 
434 

520 

508 
497 
486 
475 
464 

453 
443 
433 

o.oo  432 

43i 

430 

429 

428 

427 

426 

4-5 

424 

423 

A 

B  0 

1 

2 

3    4 

5    6 

7 

8 

9 

P  P 

a  >  &,    A=loga  —  log  6,    log  (a-\-b)  =  \oga  +  I>. 

28 


ADDITION. 

A 

B  0 

1 

2 

3 

4 

5  |  6 

7 

8 

9 

p  p 

2.0 

o.oo  432 

422 

413 

403 

394 

385 

377 

36S 

360 

352 

9 

8 

i 

344 

336 

328 

321 

313 

306 

299 

293 

286 

280 

I 

0.9 

0.8 

2 

273 

267 

261 

255 

249 

244 

238 

233 

227 

222 

2 

I.f 

5 

1.6 

3 

217 

212 

207 

203 

198 

194 

189 

185 

181 

177 

3 

2.' 

? 

2.4 

4 

173 

169 

165 

161 

157 

154 

150 

147 

144 

140 

4 

3-6 

3-2 

6 

137 
109 

134 
1  06 

131 
104 

128 

102 

125 
099 

122 

097 

119 

095 

117 
093 

114 

091 

III 

089 

6 

4-5 
5-4 
63 

4.0 

4.8 
5.6 

7 

087 

085 

083 

081 

079 

077 

071  5 

074 

072 

070 

8 

7  2 

64. 

8 

069 

067 

066 

064 

063 

06  1 

060 

059 

057 

056 

9 

8.1 

7.2 

9 

055 

053 

052 

051 

050 

049 

048 

047 

045 

044 

3.0 

o.oo  043 

042 

041 

041 

040 

039 

038 

037 

036 

035 

7 

6 

5 

i 

034 

034 

033 

032 

031 

031 

030 

029 

029 

028 

I  0.7 

0.6 

0.5 

2 

027 

927 

026 

026 

025 

024 

024 

023 

023 

O22 

2  1.4 

1.2 

I.O 

3 

O2  2 

021 

021 

O2O 

O2O 

019 

019 

019 

018 

018 

3  2.1 

1.8 

i-5 

4 

017 

017 

017 

016 

016 

015 

015 

015 

014 

014 

4  2.8 

2.4 

2.0 

5 

014 

013 

013 

013 

013 

OI2 

012 

OI2 

on 

on 

5  3-5 

3-0 

2-5 

6 

on 

on 

OIO 

OIO 

OIO 

OIO 

OIO 

009 

009 

009 

6  4.2 

3-6 

3-0 

7 
8 

009 
007 

008 
007 

008 
007 

008 
006 

008 

006 

008 

006 

008 

006 

007 
006 

007 

006 

007 

006 

7  4-9 
8  5-6 

4.2 
4.8 

3-5 
4.0 

9 

005 

005 

005 

005 

005 

005 

005 

005 

005 

004 

9  o-3 

b-4 

4-5 

4.0 

0.00004 

004 

004 

004 

004 

004 

004 

004 

004 

004 

• 

i 

003 

003 

003 

003 

003 

003 

003 

003 

003 

003 

I 

* 
0.4 

0 

o.i 

2 

003 

003 

003 

003 

OO2 

OO2 

002 

OO2 

002 

OO2 

2 

08 

0.6 

3 

OO2 

OO2 

OO2 

002 

002 

002 

OO2 

OO2 

OO2 

002 

3 

1.2 

0.9 

4 

OO2 

OO2 

OO2 

002 

002 

OO2 

002 

OOI 

OOI 

OOI 

4 

1.6 

1.2 

5 

001 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

5 

2.0 

I- 

5 

6 

001 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

6 

2.4 

I. 

3 

7 

001 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

7 

2.8 

2.1 

8 

001 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

8 

3-2 

2.4 

9 

OOI 

OOI 

OOI 

OOI 

ooo 

ooo 

000 

ooo 

ooo 

000 

9 

3-6 

2.7 

5.0 

o.ooooo 

000 

000 

ooo 

ooo 

000 

ooo 

ooo 

ooo 

000 

A 

B  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

p 

P 

a>&,     A  =  loga  —  log  6,     log  (a  +  &)  =  log  a  -f  -B. 

The  above  table  of  Addition  Logarithms  is  based  on  the  identity 


log  a  +  log 


The  argument  A  is  log  — ,  and  the  function  B  is  log  I          a  j,  conse- 

\         bj 


quently 


log(a-f  6)  =  loga-f  B. 


SUBTRACTION. 


A 

B  0 

1 

2 

3  !  4 

5 

6    7 

8  |  9 

p  P 

0.300 
301 
302 
303 
304 
305 
306 

307 
308 

309 

0.30  206 

196 

1  86 

176 

i06 

156 

146 

136 

126 

116 

9 

i  0.9 

2   1.8 

3  2.7 
4  3-6 
5  4-5 
6  54 
7  6.3 
8  7.2 
9  8.1 

8 

i  0.8 

2   1.6 

3  2-4 
4  3-2 
5  4-0 
6  4.8 

7  5-6 
8  6.4 
9  7-2 

1  06 
006 
0.29  907 
808 
710 
612 

5U 
4i7 
320 

096 
*99& 
897 
798 
700 
602 

504 
407 

3ii 

086 

*9g6 
887 

788 
690 
592 

495 
398 
301 

076 
*97& 
877 

778 
680 
582 

485 
388 
291 

066 

*966 
867 

769 
670 

573 

475 
378 
282 

056 

*956 
857 

759 
66  1 

563 
465 
369 
272 

046 
*947 
848 

749 
651 
553 
456 

359 
263 

036 

*937 
838 

739 
641 
543 
446 
349 
253 

026 

*927 
828 

729 
631 
534 
436 
340 
243 

016 

*9!7 

818 

719 
621 
524 
427 
330 
234 

0.310 

3" 
312 

3i3 
3i4 
3i5 
316 

3i7 
318 

319 
0.320 

321 
322 
323 
324 
325 
326 

327 

328 

329 
0.330 

33i 
332 
333 
334 
335 
336 

337 
338 
339 

0.340 

34i 
342 
343 
344 
345 
346 

347 
348 
349 
0.350 

224 

215 

205 

195 

186 

176 

167 

157 

147 

138 

128 

033 
0.28938 

844 
750 
656 

563 
470 

377 

119 
024 
929 
834 
740 
647 

553 
461 
368 

109 
014 
919 

825 
73i 
637 
544 
451 
359 

100 

005 
910 

815 
721 
628 

535 
442 
350 

090 

*995 
900 

806 
712 
619 

525 
433 
341 

08  1 
*986 
891 

797 
703 
609 

5i6 
424 
33i 

071 

*97& 
881 

787 
693 
600 

507 
414 
322 

062 

*9^7 
872 

778 
684 

591 
498 
405 
313 

052 

*957 
862 

768 

675 

581 

488 
396 
304 

043 
*948 

853 

759 
665 

572 

479 

387 
29! 

285 

276 

267 

258 

249 

240 

.  230 

221 

212 

203 

194 
103 

OI2 

0.27921 
831 
742 

653 

564 

475 

185 
094 
003 
912 
822 
733 
644 
555 
466 

176 
084 
*994 
903 
8i3 
724 

635 
546 

458 

1  66 

075 
*985 
894 
804 
715 
626 
537 
449 

157 
066 

*976 
885 
796 
706 
617 
528 
440 

148 
057 
*967 
876 
787 
697 
608 
519 
43i 

139 
048 

*958 
867 
778 
688 

599 
5ii 
422 

130 
039 

*948 
858 
769 
679 

590 
502 
414 

121 

030 

*939 

849 
760 
670 

581 
493 
405 

112 

021 

*930 
840 

75i 
66  1 

573 
484 
396 

387 

378 

370 

361 

352 

343 

335 

326 

3i7 

308 

300 

212 

125 
039 
0.26953 
867 

781 
696 

611 

291 
204 
H7 
030 

944 

858 

773 
688 
603 

282 

J95 
1  08 

021 

935 
850 

764 
679 
595 

273 
186 

099 
013 
927 
841 
756 
671 
586 

265 

177 
091 

004 
918 

832 

747 
662 

578 

256 
169 
082 

*99& 
910 

824 

739 
654 
569 

247 
1  60 

073 

*987 
901 

8i5 
730 
645 
56i 

238 
151 
065 

*978 
892 
807 
722 
637 
552 

230 

143 
056 

*970 

884 
798 

713 

628 

544 

221 
134 
047 

*96i 

875 
790 

705 
620 

535 

527 

519 

5io 

502 

493 

485 

477 

468 

460 

45i 

443 
359 
276 

193 
in 
028 
0.25  946 
865 
784 

435 
351 
268 

185 

IO2 
020 

938 

857 

775 

426 

343 
259 

177 
094 

OI2 

930 

849 
767 

418 
334 
251 
1  68 
086 
004 
922 
840 
759 

410 
326 
243 
1  60 
078 
*995 
914 
832 
75i 

401 
3i8 
235 
152 
069 
*987 
906 
824 
743 

393 
309 
226 

144 
06  1 
*979 
897 
816 
735 

384 
301 

218 

135 
053 
*97i 
889 
808 
727 

376 
293 

2IO 

127 
045 

*9^3 
881 
800 
719 

368 
284 

201 
119 
036 

*955 
873 
792 
711 

0.25  703 

695 

687 

678 

670 

662 

654 

646 

638 

630 

A 

B  0   1    2 

3 

4 

5 

6    7 

8 

9 

P  P 

a  >  6.        Put   x  =  log  a — log  6. 

Ifa?>.8,       then       x  —  A       and       log  (a  —  6)  =  log  a  —  B. 
If  x<  .3,       then       x  =  B       and       log  (a  —  6)  =  loga— A. 


30 


SUBTRACTION. 

A 

B  0 

1 

2 

3 

4 

5 

6 

7 

8 

y 

P  P 

0.350 

0.25  703 

695 

687 

678 

670 

662 

654 

646 

638 

630 

9 

0.9 

2.7 
3-6 
4-5 
54 
6.3 

7-2 

8.1 

8 
0.8 
1.6 
24 
3-2 
4.0 
4.8 
5.6 
6.4 
7.2 

7 
0.7 
1.4 

2.1 

2.8 

3.5 

4.2 

4.9 

5.6 
6.3 

6 

0.6 

1.2 

1.8 

2-4 

3-0 
3-6 
4-2 
4-8 
54 

35i 
352 
353 
354 
355 
356 

357 
358 
359 
0.360 

361 
362 
363 
364 
365 
366 

367 

368 

369 
0.370 

37i 
372 
373 
374 
375 
376 

377 
378 
379 
0.380 

38i 
382 
383 
384 
385 
386 

387 
388 

389 
0.390 

622 
542 
462 

382 
303 
224 

146 
067 

0.24989 

614 
534 
454 
374 
295 
216 

138 
060 

982 

606 
526 
446 

367 
287 
209 

130 
052 
974 

598 
518 
438 

359 
279 

201 
122 
044 

966 

590 
5io 
430 

351 
272 

193 
114 
036 
958 

582 
502 
422 

343 
264 

185 
106 
028 
95i 

574 
494 
414 

335 
256 
177 
099 

021 

943 

566 
486 
406 

327 
248 
169 
091 
013 
935 

558 
4/8 
398 

319 
240 
161 
083 
005 
927 

550 
4/0 
390 

3ii 

232 
154 
075 
*997 
920 

i 

2 

3 
4 

6 

7 

912 

904 

896 

889 

881 

873 

865 

858 

850 

842 

83! 
758 
681 

604 

528 
453 

377 
302 
227 

827 
750 
673 
597 
521 
445 
370 
295 

220 

819 
742 
666 

589 
513 
438 
362 
287 

212 

811 

734 
658 

582 
506 
430 

355 
280 
205 

804 

727 
650 

574 
498 
422 

347 
272 

197 

796 
719 
643 
566 
490 
415 
340 
265 
190 

788 
711 
635 

559 
483 
407 

332 
257 
182 

781 
704 
627 

55i 
475 
400 

325 
250 
175 

773 
696 
620 

544 

468 
392 

317 
242 
1  68 

765 
688 
612' 

536 
460 
38! 
310 
235 
1  60 

8 
9 

i 

2 

3 
4 

6 

8 
9 

i 

2 

3 
4 

6 

8 
9 

i 

2 

3 
4 

5 
6 

8 
9 

153 

145 

138 

130 

123 

116 

108 

IOI 

093 

086 

078 
004 
0.23  931 

857 
784 
712 

639 

567 
495 

071 

*997 
923 
850 

777 
704 

632 
560 

488 

064 

*990 
916 

843 
770 
697 
625 
553 
481 

056 

*982 
909 

836 

763 
690 

617 

545 
474 

049 
*975 
901 

828 

755 
683 

610 
538 
466 

041 

*968 
894 

821 
748 
675 
603 
53i 
459 

034 
*9&o 
887 

814 
74i 
668 

596 
524 
452 

027 

*953 
879 

806 

733 
661 

589 
517 
445 

019 
*946 
872 

799 
726 

654 
58i 
509 

438 

012 

*938 
865 

792 

719 
646 

574 
502 

43i 

423 

416 

409 

402 

395 

388 

38i 

373 

366 

359 

352 
281 

210 
140 
069 
000 
O.22  930 
860 
791 

345 
274 
203 

133 
062 

*993 
923 

853 

784 

338 
267 
196 

126 
055 
*986 
916 
847 
777 

33i 
260 
189 
119 
048 
*979 
909 
840 
771 

324 
253 
182 

112 
O4I 

*972 
902 
833 
764 

317 
246 

175 

105 
034 
*965 
895 
826 
757 

309 
238 
1  68 
098 
027 
*958 
888 
819 
750 

302 
231 
161 

091 
020 
*95i 
881 
812 
743 

295 
224 

154 
083 
013 
*944 

874 
805 
736 

288 
217 
147 
076 
006 
#937 
867 
798 
729 

722 

716 

709 

702 

695 

688 

681 

674 

667 

66  1 

39i 
392 
393 
394 
395 
396 
397 
398 
399 

654 

585 
517 
450 
382 
315 
248 

181 
114 

647 
579 
5ii 
443 
375 
308 

241 

174 
108 

640 

572 
504 
436 
369 
301 

234 
168 

IOI 

633 
56! 
497 

429 
362 

295 
228 
161 
094 

626 

558 
490 

422 

355 
288 

221 

154 
088 

620 

55i 
483 
416 
348 
281 

214 
148 
08  1 

613 
545 
477 

409 
342 
274 
208 
141 
075 

606 
538 
470 

402 

335 
268 

201 

134 
068 

599 
531 
463 

395 
328 
261 
194 
128 
06  1 

592 
524 
456 
389 
321 
254 
188 

121 
05§ 

0.400 

.22048 

041 

03! 

028 

022 

015 

008 

002 

»995 

0989 

A 

B  0 

1 

2 

3 

4 

5 

6 

7 

8 

B 

P  P 

a  >  6.   Put  x  =  log  a  —  log  6. 
Ifa;>.3,   then   x  =  A   and   log  (a—  6)  =  log  a—  B. 
Ifx<.3,   then   x  =  B   and   log  (a  —  6)  =  log  a  —  A. 

31 


SUBTRACTION. 

A 

B  0 

1    2 

3    4 

5 

678 

9 

P  P 

0.400 

401 
402 

403 
404 

405 
406 

407 
408 
409 

0.410 

411 
412 
4I31 

414 

4i5 
416 

4i7 
418 
419 

0.420 

421 
422 
423 
424 
425 
426 

427 
428 
429 

0.430 

43i 
432 
433 
434 
435 
436 

437 
438 
439 
0.440 

44i 
442 

443 
444 
445 
446 

447 
448 

449 
0.450 

O.22  048 

041 

035 

028 

022 

015 

008 

002 

*995 

*989 

i 

2 

3 
4 

6 

8 
9 

i 

2 

3 
4 

6 

8 
9 

i 

2 

3 
4 
5 
6 

8 
9 

7 
0.7 
1.4 

2.1 

2.8 

3.5 
4.2 
4.9 

5.6 
6.3 

6 

0.6 

1.2 

1.8 

2.4 
3.0 

3-6 
4.2 

4.8 
54 

5 

0.5 

I.O 

i.5 

2.0 
2-5 

3-0 
3.5 
4.0 

4-5 

O.2I  982 
916 
851 
786 
721 
656 

591 
527 
463 

975 
910 
844 

779 
7H 
649 

585 
521 
456 

969 
9°3 
838 

772 
708 
643 
578 
514 
450 

962 
897 
831 
766 
701 
636 

572 
508 
444 

956 
890 

825 

759 
695 
630 

565 
501 
437 

949 
884 
818 

753 
688 
623 

559 
495 
431 

943 

877 
812 

746 
682 
617 

553 

488 

425 

936 
870 
805 
740 
675 

611 

546 
482 
418 

929 
864 
799 

733 
669 
604 

540 
476 
412 

923 

857 
792 

727 
662 

598 
533 
469 
405 

399 

393 

386 

380 

374 

367 

361 

355 

348 

342 

336 
272 
209 
146 
084 
02  1 
0.20  959 
897 
836 

329 
266 
203 
140 
077 
015 

953 
891 
829 

323 
260 

197 

134 
071 
009 

947 

885 
823 

317 
253 
190 

127 
065 
003 
941 
879 
817 

310 
247 
184 

121 
059 

*996 
934 

873 
811 

304 
241 

178 

H5 
052 

*99° 
928 
866 

805 

298 

234 
171 

109 
046 
*9S4 
922 
860 
799 

291 
228 
165 

IO2 
040 

*978 
916 

854 
793 

285 

222 

J59 
096 
034 
*972 
910 
848 
786 

279 
215 
153 
090 
028 
*965 

903 

842 
780 

774 

768 

762 

756 

750 

743 

737 

731 

725 

719 

713 
652 

59i 

'  53i 
470 
410 

350 
291 
231 

707 
646 

585. 

525 
464 
404 

344 
285 
225 

701 
640 
579 

5i8 
458 
398 
338 

279 
219 

69! 
634 
573 
512 
452 
392 
332 
273 
213 

688 
628 
567 
506 
446 
386 

326 
267 
207 

682 
621 
56i 
500 
440 
380 
320 
261 

2OI 

676 
615 
555 
494 
434 
374 
314 
255 
196 

670 
609 

549 
488 
428 
368 
308 

249 
190 

664 
603 
543 

482 
422 
362 
302 

243 
184 

658 
597 
537 
476 
416 
356 

297 
237 

178 

172 

1  66 

1  60 

154 

148 

142 

136 

131 

125 

119 

H3 

054 
0.19  996 

937 
879 
821 

763 
706 
648 

107 

048 
990 

93i 

873 
815 
758 
700 
643 

IOI 

042 
984 
926 
867 
809 

752 
694 
637 

095 
037 
978 
920 
862 
804 

746 
689 
631 

089 
031 
972 
914 

856 
798 
740 
683 
626 

083 
025 
966 
908 

850 
792 

735 
677 
620 

078 
019 
960 
902 
844 
786 

729 
671 
614 

072 
013 
955 
896 

838 
78i 

723 
666 
608 

066 
007 
949 
891 

833 

775 

717 
660 
603 

060 

001 

943 

885 
827 
769 
712 
654 
597 

59i 

586 

580 

574 

569 

563 

557 

552 

546 

540 

534 
478 
421 

36! 
309 
253 
197 
142 
087 

529 
472 
416 

359 
303 
247 
192 
136 
081 

523 
466 
410 

354 
298 
242 

186 

131 
076 

5i7 
461 
404 

348 
292 
236 
181 
125 
070 

512 
455 
399 
343 
297 
231 

175 

120 

064 

506 
450 
393 

337 
281 
225 
170 
114 
059 

500 
444 
387 
33i 

2/5 

220 

164 
109 
053 

495 
438 
382 

326 
270 
214 

158 
103 
048 

489 
433 
376 
320 
264 
208 

153 
098 
042 

*988 

483 
427 
37i 
315 
259 
203 

147 
092 

037 

,.982 

0.19031 

026 

020 

015  009 

004 

K999 

x-993 

A 

B  0 

1 

2 

3 

4 

5 

6 

7 

8    9 

P  P 

a  >  b.    Put  x  =  log  a  —  log  b. 
Ifx>.3,   then   x—A   and   log  (a—  6)  =  log  a  —  B. 
Jf#<.3,    then   x  =  B   and   log  (a  —  6)  =  log:  a  —  A. 

SUBTRACTION. 

A 

13  0 

1 

2    3  |  4 

o 

G 

7    8    9 

P  P 

0.450 

452 
453 
454 
455 
456 

457 
458 
459 
0.460 

461 
462 
463 
464 

465 
466 

467 
468 
469 

0.470 

47* 
472 

473 
474 
475 
476 

477 
478 
479 
0.480 

481 

482 
483 
484 
485 
486 

487 
488 
489 

0.490 

0.19031 

026 

020 

015 

009 

004 

*999 

*993 

*9SS 

*982 

6 

i  0.6 

2   1.2 

3  1.8 
4  2.4 
5  3-0 
6  3.6 

7  4-2 
8  4.8 
9  5-4 

5 

i   0.5 

2    1.0 

3   1.5 

4    2.0 

5   2.5 
6   3.0 

7   3-5 
8   4.0 

9   4-5 

4 

i  0.4 

2   0.8 

3  1.2 
4  1.6 
5  2.0 
6  2.4 
7  2.8 
8  3.2 
9  3-6 

0.18977 
922 
867 

813 
759 
705 
651 

598 
544 

97i 
916 
862 
808 

754 
700 

646 

592 
539 

966 
911 
856 
802 

748 
694 

641 

587 
534 

960 
905 
851 
797 
743 
689 

635 

582 
528 

955 
900 
846 
791 

737 
683 

630 
576 
523 

949 
895- 
840 

786 

732 
678 

624 
571 

944 
889 

835 
781 
727 
673 
619 
566 
512 

938 
884 
829 

775 

667 

614 
560 
507 

933 
878 
824 

770 

662 

608 

555 
502 

027 
873 
818 

764 
710 
657 
603 
550 
496 

491 

486 

481 

475 

470 

4^5 

459 

454 

449 

443 

438 
385 
333 
280 
228 
176 
124 
072 

021 

433 
380 
328 

275 
223 
171 
119 
067 
016 

428 
375 
322 
270 
218 
166 
114 
062 
on 

422 
370 
317 
265 

212 

160 

109 

057 
006 

417 
364 
312 

259 
207 

155 
103 
052 

000 

412 
359 
307 

254 
202 
150 
098 

047 
*995 

406 

354 
301 

249 
r97 
145 

093 
042 

*99° 

401 

349 
296 

244 
192 
140 
088 
036 

396 
343 
291 

239 
186 

135 
083 
031 

338 
286 

233 
181 
129 
078 
026 
*975 

0.17970 

964 

959 

954 

949 

944 

939 

934 

929 

924 

918 
867 
8I7 
766 
7I6 
665 

615 

565 
515 

913 
862 
812 
761 

660 
610 
560 
5" 

908 
857 
807 

756 
706 

655 
605 
555 
506 

903 
852 
801 

700 
650 
600 
550 

898 
847 
796 

746 
695 
645 

595 
545 
496 

893 
842 
791 

690 
640 

590 
540 
491 

888 

837 
786 

736 
685 
635 
585 
535 
486 

883 
832 

731 
680 
630 
580 
530 
481 

878 
827 
776 
726 
675 
625 

575 
525 
476 

873 
822 

771 
721 
670 
620 

5/0 
520 
471 

466 

461 

456 

45i 

446 

441 

436 

43i 

426 

421 

4l6 
367 
318 
269 
220 
172 
123 

075 
027 

412 
362 
313 
264 
216 
167 
119 
070 
022 

407 
357 
308 

259 

211 
162 
114 
066 

018 

402 
352 
303 

255 
206 

157 
109 
061 
013 

397 
348 
299 
250 

2OI 

153 
104 
056 
008 

392 
343 
294 

245 
196 

148 
099 

051 
003 

387 
338 
289 

240 
191 
143 

09! 
046 

*998 

382 
333 
284 

235 
186 
138 
090 
042 
*994 

377 
328 

279 
230 
182 

133 
085 
037 
*989 

3/2 

323 

274 
225 

177 

128 

080 
032 

0.16979 

974 

970 

965 

960 

955 

95i 

946 

941 

93f> 

491 
492 
493 
494 
495 
496 

497 
498 

499 

931 
884 
836 

789 
742 

695 
648 
602 

555 

927 
879 

832 
784 

737 
690 

644 

597 

551 

922 
874 
827 
780 

733 
686 

639 

592 
546 

917 

870 
822 

775 
7=8 
68  1 

634 
588 
541 

912 
865 

818 
770 
723 
676 

630 

533 
5.17 

908 
860 
813 
766 
719 
672 

625 

5/8 

903 

855 
808 

761 

667 
620 
574 
527 

898 
851 
803 

756 
709 
662 
616 
569 
5=3 

893 
846 

799 

75i 
704 
658 
611 

889 
841 
794 

747 
700 

653 
606 
560 
513 

0.800 

0.16  509 

504 

500 

49§ 

49° 

486 

481 

477 

472 

467 

A 

13  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P  P 

a  >  b.   Put  x  =  log  a  —  log  6. 
Ifoj>.3,   then   x-A   and   log  (a  —  6)  =  log  a—  B. 
Tf.T<.3,   then   x  =  J)   and   log  (a  —  6)  =  loga—  A. 

SUBTRACTION. 

A 

B  0 

1 

2  |  8 

4 

5 

6 

7 

8 

9 

P  P 

0.50 

o.i  6  509 

463 

4i7 

371 

325 

280 

234 

189 

144 

099 

5I 

054 

009 

*965 

*92i 

*876 

*832 

*788 

*745 

*70i 

*657 

46 

45 

44 

43 

52 

0.15  614 

57i 

528 

485 

442 

400 

357 

315 

273 

230 

i 

4.6 

4-5 

4-4 

4-3 

53 

189 

147 

105 

064 

022 

*98i 

*94o 

•*899 

*858 

*8i7 

2 

9.2 

9.0 

8.8 

8.6 

54 

55 

0.14777 

378 

736 
339 

696 
300 

656 
261 

616 

222 

576 
183 

536 
145 

496 
106 

457 
068 

417 
030 

3 
4 

13-8 
18.4 

13.5 
1  8.0 

13-2 
17.6 

12.9 
17.2 

56 

57 
58 
59 

0.13  992 
617 
255 

0.12  903 

954 

581 
219 
869 

916 

544 

184 

834 

878 

507 
148 
800 

840 

471 

766 

803 

435 
078 
732 

766 

398 
043 

698 

728 
362 
008 
664 

691 
326 

*973 
630 

654 
291 

*938 
596 

5 
6 

8 
9 

23.0 
27.6 
32.2 
36.8 
41.4 

22.5 
27.0 

3i-5 
36.0 

40.  * 

22.0 
26.4 
30.8 
35-2 

21.5 
25.8 
30.1 
34-4 

18.7 

0.60 

563 

529 

496 

463 

429 

396 

363 

330 

298 

265 

61 

232 

200 

1  68 

135 

103 

071 

039 

007 

*975 

*944 

42 

41 

40 

39 

62 

o.n  912 

880 

849 

818 

786 

755 

724 

693 

663 

632 

i 

4.2 

4.1 

4.0 

3-9 

63 

60  1 

571 

540 

510 

479 

449 

419 

389 

359 

329 

2 

8.4 

8.2 

8.0 

7.8 

« 

12.6 

12.1 

I2.O 

II  7 

64 
65 
66 

299 
OO7 

o.io  722 

270 

*978 
694 

240 

*949 
667 

211 

639 

181 
*892 
611 

152 
*864 
583 

123 

*835 
556 

094 
*528 

065 

*779 
SGI 

036 
474 

j 

4 

5 
6 

1  6.8 

2I.O 

25.2 

*  ~o 
16.4 
20.5 
24.6 

16.0 

2O.O 

24.0 

•  / 
15-6 
19.5 
23.4 

67 

446 

419 

392 

365 

338 

312 

285 

258 

231 

205 

7 

294 

28.7 

28.0 

27-3 

68 

178 

152 

126 

100 

073 

047 

02  1 

*995 

*97o 

*944 

8 

33-6 

32.8 

32.0 

31.2 

69 

0.09  918 

893 

867 

842 

816 

791 

766 

740 

715 

690 

9 

37-8 

36.9 

36.0 

35-1 

0.70 

665 

640 

616 

591 

566 

542 

517 

493 

468 

444 

7i 

420 

395 

37i 

347 

323 

299 

275 

252 

228 

204 

38 

_  Q 

37 

36 

_  f_ 

35 

72 
73 

181 

0.08  949 

157 
926 

134 
9°3 

no 

880 

087 
858 

064 

835 

041 
8i3 

018 
790 

*995 
768 

*972 
745 

i 

2 

3-8 
7-6 

1  1.  4 

3-7 
7.4 
ii.  i 

3-6 
7-2 
10.8 

3-5 
7.0 

74 

723 

701 

679 

657 

635 

613 

591 

569 

547 

525 

A  i  •<-(. 

1^.2 

14.8 

14.4 

14.0 

75 

504 

482 

461 

439 

418 

396 

375 

354 

333 

311 

5" 

*  ?*m 

18  5 

18.0 

17.  c 

76 

290 

269 

248 

228 

207 

186 

165 

145 

124 

103 

6 

22.8 

22.2 

21.6 

/  "Z3 

21.0 

77 

083 

063 

042 

022 

002 

*98  1 

#96  1 

*94i 

*921 

#9O1 

7 

26.6 

25.9 

25.2 

24.5 

78 

0.07  881 

861 

842 

822 

802 

-782 

763 

743 

724 

704 

S 

30.4 

29.6 

28.8 

28.0 

79 

685 

666 

646 

627 

608 

589 

570 

532 

513 

9 

34-2 

33-3 

32.4 

3L5 

0.80 

494 

475 

456 

438 

419 

401 

382 

363 

345 

327 

OA 

qq 

qo 

q-j 

81 

308 

290 

272 

253 

235 

217 

199 

181 

163 

145 

I 

ot 

34 

oo 

3-3 

9m 

3-2 

ol 

3.1 

82 

127 

no 

092 

074 

056 

039 

021 

004 

#986 

*969 

2 

6.8 

6.6 

6.4 

6.2 

83 

0.06  951 

934 

917 

9OO 

882 

865 

848 

831 

814 

797 

3 

10.2 

9.9 

9.6 

9-3 

84 

780 

763 

747 

730 

713 

696 

680 

663 

647 

630 

4 

13.6 

13.2 

12.8 

12.4 

85 

614 

597 

581 

564 

548 

532 

516 

499 

483 

467 

5 

17.0 

16.5 

16.0 

15-5 

86 

45i 

435 

419 

403 

387 

372 

356 

340 

324 

309 

6 

20.4 

19.8 

19.2 

18.6 

87 

293 

278 

262 

247 

231 

216 

2OO 

185 

170 

155 

7 

23.8 

23.1 

22.4 

21.7 

88 

139 

124 

109 

094 

079 

064 

049 

034 

019 

004 

8 

27.2 

26.4 

25.6 

24.8 

89 

0.05  989 

975 

960 

945 

931 

916 

901 

887 

872 

858 

9 

30.6 

29.7 

28.8 

27.9 

0.90 

844 

829 

815 

800 

786 

772 

758 

744 

730 

715 

on 

29 

28 

27 

91 

701 

687 

673 

659 

646 

632 

618 

604 

590 

577 

i 

9J\J 

3-0 

t**7 

2-9 

0O 

28 

Lt  I 

2.7 

92 

563 

549 

536 

522 

509 

495 

482 

468 

455 

441 

2 

6.0 

5-8 

5.6 

5-4 

93 

428 

415 

401 

388 

375 

362 

349 

336 

323 

310 

3 

9.0 

8.7 

8.4 

8.1 

94 

297 

284 

271 

258 

245 

232 

219 

207  i  194 

181 

4 

I2.O 

11.6 

II.  2 

10.8 

95 

169 

156 

143 

131 

118 

106 

093 

081 

069 

056 

5 

15-0 

14-5 

14.0 

13-5 

96 

044 

032 

019 

007 

*995 

^983 

*95S 

*946 

#934 

6 

1  8.0 

17.4 

16.8 

16.2 

97 

0.04  922 

910 

898 

886 

874 

863 

^851 

839 

827 

815 

7 

21.0 

20.3 

19.6 

18.9 

98 

804 

792 

780 

769 

757 

746 

734 

723 

711 

700 

8 

24.0 

23.2 

22.4 

21.6 

99 

688 

677 

666 

654 

643 

632 

620 

609 

598 

587 

9 

27.0 

26.1 

25.2 

24.3 

1.00 

0.04  576 

56! 

554 

543 

532 

521 

5io 

499 

488 

477 

A 

B  0 

1 

2    3 

4 

5 

6 

m 
i 

8 

9 

P  P 

a  >  b.   Put   a;  =  log  a  —  log  6. 

If  x  >  .  3,   then   x  —  A   and   log  (a  —  &)=loga  —  B. 

Ifa3<.3,   then   x  =  B   and   log  (a-  &)=loga  —  A. 

SUBTRACTION. 

A 

B  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P  P 

1.00 

0.04  576 

565 

554 

543 

532 

521 

5io 

499 

488 

477 

i 

•2 

3 
4 

5 

() 

7 

S 

9 

i 

3 
4 
5 
6 

7 

S 

9 
i 

2 

3 
4 
5 
6 

7 

S 

9 
I 

2 

3 
4 

6 
8 

26 

2.6 

5.2 
7.8 
10.4 
13.0 
15.6 
18.2 

20.8 

23.4 

22 

2.2 

4-4 

6.6 

8.8 

1  1.0 

13.2 
154 
iy.C 
19.8 

18 
1.8 

3-6 
5-4 
7-2 
9.0 
10.8 

12.6 

14.4 
16.2 

14 

1.4 

2.8 

4.2 

5.6 
7.0 
8.4 
9.8 

II.  2 
12.6 

9 

0.9 
1.8 
2.7 
3-6 
4.5 
5-4 
6.3 
7-2 
8.1 

25 

2.5 
5-o 
7-5 

1  0.0 

12.5 
15.0 
17.5 

2O.O 
22.5 

21 

2.1 
4.2 

6-3 
8.4 
10.5 

I2.C 
14.1 

16.8 

I8.Q 

17 
i.7 
34 

5-i 
6.8 

8-5 

IO.2 
II.9 
13.6 
15.3 

13 

i-3 

2.6 

3.9 
5.2 
6-5 
7.8 
9.1 
10.4 
11.7 

8 
0.8  o 
1.6  i 
2.4  2 

3-2  2 

4-0  3 
4.8  4 
5-64 
64  5 
7.2  C 

24 

2.4 
4.8 
7-2 
9.6 

12.0 
144 

16.8 
19.2 

21.6 

20 

2.0 
4.0 

6.0 
8.0 

IO.O 
I2.O 
14.0 

16.0 
1  8.0 

16 

1.6 
3-2 
4.8 
6.4 
8.0 
9.6 

II.  2 

12.8 

14.4 

12 

1.2 

24 

3.6 

4.8 
6.0 

7-2 

8.4 
9.6 
10.8 

7  e 

.7  o. 
4  i. 
.1  i. 

.8  2. 

.5  3. 

•2  3- 
•9  4- 
.(>  4. 
•3  5- 

23 

2.3 
4.6 
6.9 
9.2 
ii.  5 
13-8 
16.1 
18.4 
20.7 

19 

1-9 
3-8 
5-7 
7-6 
9-5 
11.4 
13-3 
15-2 
17.1 

15 

1.5 

3-0 

4-5 

6.0 

7-5 
9.0 
10.5 

I2.O 
13.5 

11 

I.I 

2.2 

3-3 
4-4 
5-5 
6.6 

7.7 
8.S 
9.9 

5 

5  0.5 

2  1.0 

B  1.5 
4  2.0 

3  2.5 
63.0 

2  3.5 

B  4.0 
4  4-5 

01 

02 
03 
04 

05 
06 

07 
08 

09 

466 
359 
255 
153 
054 
0.03  958 

863 

771 
682 

455 
349 
245 
143 
044 
948 

854 
762 

673 

444 
338 
234 
133 
035 
938 

845 
753 
664 

434 
328 
224 
123 
025 
929 

835 
744 
655 

423 
3i7 
214 

113 
015 
920 

826 
735 
647 

412 

307 
204 

103 
006 
910 

817 
726 
638 

402 
296 
194 

093 
*996 
901 

808 

717 
629 

39  ! 
286 

183 
084 
*9S6 
891 

799 
708 
620 

380 
275 
173 
074 
*977 
882 

790 
700 
612 

370 
265 
163 
064 
*967 
873 
78i 
691 
603 

1.10 
ii 

12 
13 
14 
15 

16 

17 

18 

19 
1.20 

21 
22 
23 
24 

25 
26 

27 

28 
29 

1.30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 

1.40 

4i 
42 

43 
44 
45 
46 

47 
48 

49 
1.50 

594 

586 

577 

569 

560 

552 

543 

535 

526 

5i8 

509 
426 

345 
266 
189 
114 
040 

0.02  969 
899 

501 
418 
337 
258 
181 
106 

033 
961 
892 

492 
410 
329 
250 

174 

099 

026 

954 

885 

484 
402 
321 

243 

166 
091 
018 

947 

878 

476 
393 
313 
235 
159 
084 

on 
940 

871 

467 
385 
305 
227 
151 
077 
004 

933 

864 

459 
377 
297 
219 
143 
069 

*997 
926 

858 

45i 
369 
289 

212 
I36 
062 

*990 
919 

851 

443 
361 
282 
204 
128 
055 

*983 
912 

844 

434 
353 
274 
196 

121 

047 

*976 
906 

837 

830 

824 

817 

810 

804 

797 

790 

784 

777 

77i 

764 
699 
636 

574 
5U 
455 
397 
34i 
286 

757 
693 
629 

568 
508 
449 
392 
336 
281 

75i 
686 
623 
562 
502 
443 
386 
330 
276 

744 

680 

617 

556 
496 
437 
380 

325 
270 

738 
674 
611 

550 
49° 
432 

375 
319 
265 

73i 
667 
605 

544 
484 
426 

369 
3H 
260 

725 
66  1 

598 
538 
478 
420 

363 
308 
254 

718 
655 
592 

532 
472 
414 

358 
303 
249 

712 
648 
586 
526 
466 
409 

352 
297 
244 

705 
642 
580 
520 
461 
403 

347 
292 
238 

233 

228 

223 

217 

212 

207 

202 

196 

191 

iS6 

181 
130 
080 

032' 
o.oi  985 
938 

893 
849 
806 

176 
125 
075 
027 
980 
934 
889 

845 
802 

171 

120 
071 
022 

975 
929 
884 
841 
798 

1  66 

H5 
066 

018 
9/i 
925 
880 
836 
794 

1  60 

no 
061 

013 
966 
920* 
876 
832 
789 

155 
105 
056 

008 
961 
916 

871 
828 

785 

150 

IOO 

051 
003 

957 
911 

867 
823 
781 

145 
095 
046 

*999 
952 
907 

862 
819 
777 

140 
090 
042 

*994 
948 
902 

858 
815 
773 

135 
085 

037 

*989 
943 
898 

854 
Sii 
768 

764 

760 

756 

752 

748 

744 

740 

736 

73i 

727 

723 
683 
644 
606 
569 
533 

497 
462 

429 

719 

679 
640 

602 

565 
529 
494 
459 
425 

715 
675 
637 

599 
562 

525^ 
490' 
456 
422 

711 
672 
633 
595 
558 
522 

487 
452 
419 

707 
668 
629 

591 
554 
518 

483 
449 
415 

703 
664 
625 

587 
55i 
5i? 
.480 
445 
412 

699 
660 
62  & 

584 
547 
5ii 
476 
442 
409 

695 
656 
618 
580 

543 
508 

473 
439 
405 

691 
652 
614 
576 
540 
504 
469 

435 
402 

687 
648 
610 

573 
536 
501 
466 
432 
399 

i 

M 

3 
4 

5 

6 

7 

S 

o.oi  396 

392 

389 

386 

383 

379 

376 

373 

370 

366 

{> 

A 

B  0 

1 

2 

3 

4 

5 

8 

7 

8 

9 

\       P   P 

a  >  b.   Put  x  =  log  a  —  log  b. 
Ifo;>.3,   then   x—  A   and   log(a—  6)=  loga  —  B. 
]fa?<.3,    then   x  —  B   and   log  (a  —  f;)  =  lo£rt  —  A. 

SUBTRACTION. 

A 

B  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P  P 

150 

o.oi  396 

392 

389 

386 

383 

379 

376 

373 

370 

366 

I 

2 

3 
4 

6 

7 
8 

9 
i 

2 

3 
4 
5 
6 

8 
9 

4 

0.4 

0.8 

1.2 

1.6 

2.O 

2.4 

2.8   - 

3-2 
3.6 

3 

0.3 
0.6 
0.9 

1.2 

i-5 
1.8 

2.1 

2.4 
2.7 

2 

0.2 

0.4 
0.6 
0.8 

I.O 
1.2 
1.4 

1.6 

1.8 

5i 

52 
53 
54 

55 
56 
57 
58 
59 

1.60 

61 
62 
63 
64 
65 
66 

67 
68 
69 

1.70 

7i 

72 
73 
74 
75 
76 

77 
78 
79 
1.80 

81 
82 
83 
84 
85 
86 

87 

88 

89 
1.90 

9i 
92 
93 
94 
95 
96 

97 
98 

99 
2.00 

363 
332 
301 
271 
242 
213 

185 
158 
131 

360 

329 
298 

268 
239 

2IO 

182 

155 
128 

357 
326 

295 
265 
236 
207 
179 
152 
126 

354 
322 
292 
262 

233 
204 

177 
150 
123 

35i 
319 
289 

259 
230 
202 

174 

147 

120 

347 
316 
286 
256 
227 
199 
171 
144 
118 

344 
313 
283 

253 
224 
196 
168 
142 
H5 

341 
310 

280 

250 

221 

193 

1  66 
139 
H3 

338 
307 

277 

247 
219 
190 

163 
136 
no 

335 
304 
274 

244 
216 

188 
1  60 

134 
107 

o.oi  105 

102 

IOO 

097 

095 

092 

089 

087 

084 

082 

079 
054 
030 

006 
0.00983 
961 

939 
917 
896 

077 
052 
028 
OO4 
981 
958 

93  (> 
915 

894 

074 
050 
025 
002 

979 

956 

934 
913 

892 

072 

047 
023 

*999 
976 

954 
932 
911 
890 

069 

045 
02  1 

*997 
974 

952 
•^30 
908 

888 

067 
042 
018 

*995 
972 
950 
928 
906 
886 

064 
040 
016 

*993 
970 

947 
926 
904 

883 

062 
037 
014 

*990 
967 

945 

923 
902 

881 

059 
035 
on 

*988 
965 
943 
921 
900 
879 

057 
033 
009 

*gS6 
963 
941 

919 

898 
877 

o.oo  875 

873 

871 

869 

867 

865 

863 

861 

859 

857 

855 
836 
816 
798 

779 
761 

744 
727 
710 

853 
834 
814 

796 

777 
760 

742 
725 

708 

851 
832 
813 

794 
776 
758 
740 
723 
707 

849 
830 
8n 

792 
774 
756 

739 

722 

705 

847 
828 
809 
790 
772 
754 

737 
720 
704 

845 
826 
807 
788 
770 
753 

735 
718 
702 

843 
824 
805 

787 
768 

751 
734 
717 
700 

841 

822 
803 

785 
767 

749 
732 
715 
699 

839 
820 
801 

783 
765 
747 
730 
713 
697 

837 

818 
799 
781 
763 
746 
728 
712 
695 

o.oo  694 

692 

691 

689 

687 

686 

684 

683 

681 

679 

678 
662 
647 
632 

618 
604 

590 
576 

563 

676 
661 
646 

631 
616 
602 

588 

575 
562 

675 
659 
644 

629 

615 
60  1 

587 
574 
56i 

673 
658 

643 
628 
614 
599 

586 
572 
559 

672 
656 
641 

626 
612 

598 

584 
57i 
558 

670 

655 
640 

625 
611 

597 

583 
570 

557 

669 

653 
638 

624 
609 

595 
582 
568 
555 

667 
652 
637 
622 

608 
594 
580 
567 
554 

665 
650 
635 
621 
606 
593 

579 
566 

553 

664 
649 
634 
619 

605 
59i 

578 
564 
55i 

i 

2 

3 

o.oo  550 

549 

548 

546 

545 

544 

543 

54i 

540 

539 

538 

525 
513 

502 
49° 
479 
468 

457 
447 

536 
524 
512 
500 
489 
478 
467 
456 
446 

535 
523 
5ii 

499 

488 

477 
466 
455 
445 

534 
522 
5io 
498 
487 
476 

465 
454 
444 

533 
520 

509 

497 
486 

474 
464 
453 
443 

531 
5^9 
507 
496 
484 
473 
462 
452 
442 

530 
518 
506 

495 
483 
472 
461 

45i 
441 

529 
517 
505 

493 
482 
471 
460 
450 
440 

528 
5i6 
504 

492 
481 
470 

459 
449 
439 

527 
5H 
503 

49  1 
480 
469 

458 
448 
437 

4 
6 

8 
9 

o.oo  436 

435 

434 

433 

432 

43i 

430 

429 

428 

427 

B  0 

1 

2 

3 

4 

•  5 

6 

7 

8    9 

P  P 

a  >  6.   Put  x  =  log  a  —  log  6. 
Ifo;>.3,   then   x  =  A   and   log  (a  —  6)=  log  a  —  B. 
Ifo?<.3,   then   os  =  S   and   log  (a  —  6)  =  log  a—  A. 

SUBTRACTION. 

A 

B    0 

1 

2 

3 

4 

5 

B 

7    [    8 

9 

p 

p 

2.0 

o.oo  436 

426 

417 

407 

398 

389 

380 

371 

363 

354 

9 

8 

i 

346 

338 

331 

323 

316 

309 

302 

295 

288 

281 

I 

0.9 

0.8 

2 

275 

269 

262 

256 

251 

245 

239 

234 

229 

223 

2 

1.8 

1.6 

3 

218 

213 

208 

204 

199 

194 

190 

1  86 

181 

177 

3 

2.7 

2.4 

4 

173 

169 

165 

162 

158 

154 

151 

147 

144 

141 

4 

3.6 

3.2 

5 

138 

134 

131 

128 

125 

123 

120 

117 

114 

112 

6 

4-5 

c    i 

4.0 

1  R 

6 

109 

107 

104 

102 

100 

097 

095 

093 

091 

089 

7 

D-4 

6.3 

4.0 

5.6 

7 

087 

085 

083 

081 

079 

077 

076 

074 

072 

070 

8 

7.2 

6.4 

8 

069 

067 

066 

064 

063 

061 

060 

059 

057 

056 

9 

8.1 

7.2 

9 

055 

053 

052 

051 

050 

049 

048 

047 

046 

044 

3.0 

o.oo  043 

042 

041 

041 

040 

039 

038 

037 

036 

035 

7 

6 

5 

i 

035 

034 

033 

032 

031 

031 

030 

029 

029 

028 

I   0.7 

0.6 

0.5 

2 

027 

027 

026 

026 

025 

024 

024 

023 

023 

022 

2 

1.4 

1.2 

1.0 

3 

022 

021 

02  1 

O2O 

O2O 

019 

019 

019 

018 

018 

3 

2.1 

1.8 

1.5 

4 

017 

017 

017 

016 

016 

015 

015 

015 

014 

014 

4 

2.8 

2.4 

2.0 

5 

014 

013 

013 

013 

013 

OI2 

OI2 

OI2 

Oil 

on 

5 

3.5 

3.0 

2.5 

6 

Oil 

on 

OIO 

OIO 

OIO 

OIO 

OIO 

009 

009 

009 

6    4.2 

3.6 

3.0 

7 

OO9 

008 

008 

008 

008 

008 

008 

007 

007 

007 

4-9 

4.2 

i  R 

3.5 

8 

007 

007 

007 

006 

006 

006 

006 

006 

006 

006 

1 

4.0 

4.0 

9 

005 

005 

005 

005 

005 

005 

005 

005 

005 

004 

9 

'3 

54 

4.5 

4.0 

O.OOOO4 

004 

004 

004 

004 

004 

004 

004 

004 

004 

i 

003 

003 

003 

003 

003 

003 

003 

003 

003 

003 

I 

0.4 

0.3 

2 

003 

003 

003 

003 

002 

OO2 

002 

OO2 

OO2 

OO2 

2 

0.8 

0.6 

3 

OO2 

002 

002 

002 

002 

OO2 

002 

002 

002 

002 

3 

1.2 

0.9 

4 

002 

002 

002 

002 

002 

OO2 

002 

OOI 

OOI 

OOI 

4 

1.6 

1.2 

5 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

5 

2.0 

I.. 

6 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

6 

2-4 

IJ 

I 

7 

OOI 

OOI 

OOI 

OOI 

OOI 

oof 

OOI 

OOI 

OOI 

OOI 

7 

2.8 

2.1 

8 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

OOI 

8 

3.2 

2.4 

9 

OOI 

OOI 

OOI 

OOI 

000 

ooo 

ooo 

ooo 

ooo 

ooo 

9 

3-6 

2.7 

5.0 

o.oo  ooo 

ooo 

000 

ooo 

ooo 

000 

000 

000 

ooo 

ooo 

A 

B    0 

1 

2 

3 

4 

5 

6 

7    |    8 

9 

p 

p 

a  >  &,             A  =  loga-logo,             log  (a-  6)  =  loga-B. 

or                  jB  =  loga  —  log  6,             log  (a  —  6)  =  log  a  —  A. 

The  above  table  of  Subtraction  Logarithms  is  based  on  the  identity 

/_2_\                         r    x     ^ 
log  (a  —  b)  =  log  1      x      1  =  log  a  —  log  (-  —  r  J, 

\«—  V 

a 
where  x=  —  . 

0 

C    x    ~\ 
The  argument  is  log  x,  and  the  function  is  log!  T). 

A  is  the  argument  and  B  the  function  when  log  x  >  .  3,  and 

B  is  the  argument  and  A  the  function  when  log  x  <  .  3. 

III 

TABLE  OF  THE   LOGARITHMS 

OP  THE 

TRIGONOMETRIC   FUNCTIONS 

FROM  0°  TO  1°  AND  89°  TO  90°  FOR  EVERY  SECOND, 

AND 

PROM  1°  TO  6C  AND  84°  TO  89°  FOR  EVERY  TEN  SECONDS. 

L  Cos      *90                L  Sin                        O°                      L  Tan             180°    *270° 

o.oo 

'  " 

0" 

1" 

2" 

3" 

4" 

5" 

6" 

7" 

8" 

9" 

10" 

ooo 
ooo 

000 
000 
000 

ooo 

o 

1C 

20 

3C 

40 
50 

5^68557 
98660 

6.  16270 

28763 
38454 

68557 
72697 
*oo779 
17694 
29836 
39315 

98660 
76476 
^02800 
19072 
30882 
40158 

#16270 
79952 
4=04730 
20409 

31904 
40985 

#28763 
83170 
$06579 
21705 
32903 
41797 

*38454 
86167 
#08351 
22964 
33879 
42594 

#46373 
88969 
#10055 
24188 
34833 
43376 

#53067 
91602 
#11694 
253/8 
3576/ 
44145 

#58866 
94085 

#13273 
26536 
36682 
44900 

#63982 
96433 
#14797 
27664 

37577 
45643 

#68557 
98660 
^16270 
28763 
38454 
46373 

50 
40 
30 

20 
IO 

o59 

ooo 
ooo 
ooo 

000 
000 
000 

ooo 
ooc 
ooc 

000 

ooc 
ooc 

0 
TO 

20 

30 
40 
50 

6.4  6373 

6.5  3067 

8866 
6.6  3982 

8557 
6.7  2697 

7090 
3683 
9406 
4462 
8990 
3090 

7797 
4291 

9939 
4936 
9418 
3479 

8492 
4890 
#0465 
5406 
9841 
3865 

9J75 
548i 
#098  5 
5870 
^0261 
4248 

9849 
6064 

*H99 
6330 
#0676 
4627 

#0512 
6639 
#2007 

6785 
*io8S 
5003 

#1165 
7207 
#2509 
7235 
#1496 
5376 

^1808 

7767 
#3006 
7680 
#1900 

5746 

#2442 
8320 
#3496 
8121 
#2300 
6112 

#3067 
8866 
#3982 
8557 
#2697 
6476 

50 
40 
30 

20 
10 

o58 

2  o 

IO 

20 

30 
40 
50 

3  o 

IO 
20 

30 
40 

50 

6476 
9952 
6.83170 
6167 
8969 
6.9  1602 

6836 
#0285 
3479 
6455 
9240 

1857 

7193 
#0615 
3786 
6742 
9509 

2110 

7548 
*0943 
4091 
7027 
9776 
2362 

7900 
^1268 

4394 
7310 

*0042 

2612 

8248 

*I59! 
4694 

7591 
#0306 
2861 

8595 
#19!! 
4993 
7870 
#0568 
3109 

8938 
#2230 

5289 

8i47 
#0829 

3355 

9278 
#2545 
5584 
8423 
*ioS8 

3599 

9616 

#2859 
5876 
8697 
#1346 

3843 

9952 
#3170 
6167 
8969 

*I602 

4085 

50 
40 
30 

20 
IO 

o  57 

ooc 
ooc 
ooc 
ooc 
ooc 
ooc 

ooo 
ooc 
ooc 
ooc 
ooc 
ooc 

4085 

'    6433 
8660 
7.00779 
2800 
4730 

4325 
6661 
8877 
0986 

2997 
4919 

4565 

6888 

9093 
1191 

3193 
5106 

4803 
7113 
9307 
1395 

3388 
5293 

5039 
7338 
9520 

1599 

3582 

5479 

5275 
756i 
9733 
1801 
3776 
5064 

5509 
7783 
9944 
2003 
3968 
5849 

5742 
8004 
#0155 
2203 
4160 
6032 

5973 
8224 
#0364 
2403 
435i 
6215 

6204 
8443 
#0572 
2602 
4541 
6397 

6433 
8660 
#0779 
2800 
4730 
6579 

50 
40 
30 

20 
10 

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L  Cos 


L  Sin 


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144 

143 

142 

141 

140 

139     138 

137 

136 

135 

134 

133 

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14.2 

14.1 

14.0 

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13-8 

13-7 

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28.8 

28.6 

28.4 

28.2 

28.0 

27.8 

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27.6 

27.4 

27.2 

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26.8 

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3 

43.2 

42.9 

42.6 

42.3 

42.0 

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41.4 

41.1 

40.8 

40.5 

40.2 

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56.8 

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53-6 

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70.5 

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85.8 

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121.5 

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131 

130 

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12.6 

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91.7 

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112.5 

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110.7 

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119 

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117 

116 

115 

114 

113 

112 

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110 

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*90°  180°  *270° 


108 

107 

106 

105 

104 

103     102 

101 

99 

98 

97 

96 

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10.8 

10.7 

10.6 

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20.8 

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20.2 

19.8 

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19.4 

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30.6 

30.3 

29-7 

29.4 

29.1 

28.8 

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42.8 

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41.6 

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40.8 

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39.2 

38.8 

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5 

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53.5 

53-0 

52-5 

52.0 

51.5 

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51.0 

50.5 

49-5 

49.0 

48.5 

48.0 

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6 

64.8 

64.2 

63.6 

63.0 

62.4 

61.8 

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60.6 

59-4 

58.8 

58.2 

57-6 

6 

7 

75-6 

74-9 

74-2 

73-5 

72.8 

72.1 

7 

71.4 

70.7 

69-3 

68.6 

67.9 

67.2 

7 

8 

86.4 

85.6 

84.8 

84.0 

83.2 

82.4 

8 

81.6 

80.8 

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78.4 

77-6 

76.8 

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9 

97-2 

96«3 

95-4 

94-5 

93-6 

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91.8 

90.9 

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88.2 

87.3 

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9 

95 

94 

93 

92 

91 

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88 

87 

86 

85 

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9.2 

9.1 

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8.9 

8.8 

8-7 

8.6 

8-5 

8.4 

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2 

19.0 

18.8 

18.6 

18.4 

18.2 

1  8.0 

2 

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17.6 

17.4 

17.2 

17.0 

16.8 

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3 

28.5 

28.2 

27.9 

27.6 

27.3 

27.0 

3 

26.7 

26.4 

26.1 

25.8 

25.5 

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4 

38.0 

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37.2 

36.8 

36.4 

36.0 

4 

35-6 

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34-8 

34-4 

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33-6 

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5 

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46.5 

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42.5 

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6 

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56.4 

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52.8 

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51-6 

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7 

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65.8 

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63.7 

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61.6 

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1  02 
246 

389 
532 
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126 
270 
413 

556 
698 

150 
294 

437 
580 
722 

461 
603 
746 

59 

58 
57 
56 
55 

833 
832 
831 

830 

829 

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24 

2.4 

4.8 

829 

828 

827 
825 
824 

5 
6 

7 
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9 

746 
887 
8.95  029 
170 
310 

769 
911 
052 
193 

333 

793 
935 
076 
216 

357 

817 
958 
099 
240 
380 

840 
982 
123 
263 
403 

864 

146 
287 
427 

887 

*029 
170 
310 

450 

54 
53 
52 

50 

828 
827 
825 

824 

823 

3 
4 
5 
6 

7 

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7.2 
9.6 

12.0 

144 

16.8 

CO  CO  CO  CO  CO 

M  to  to  to  to 

O  O  M  tO  OJ 

10 
ii 

12 
13 
14 

450 

589 
728 
867 
8.96  005 

473 
613 
752 
890 
028 

496 
636 
775 
913 
051 

520 

659 
798 

936 
074 

543 
682 
821 

959 
097 

566 
705 
844 
982 

120 

589 
728 
867 

49 

48 
47 
46 

45 

822 
821 

820 
819 

817 

9 

19.2 

21.6 

23 

817 
816 

8i5 
814 

813 

15 

16 

17 
18 

143 
280 

417 

553 
689 

1  66 

303 
440 
576 
712 

189 
326 
462 
599 

735 

212 

349 
485 
621 

757 

234 

508 
644 
780 

257 

394 
531 
667 
802 

280 
417 
553 
689 
825 

44 
43 
42 

40 

816 

815 

814 

813 

812 

2 

3 
4 

6 

2.3 
4.6 
6.9 
9-2 

U-5 

13-8 

812 
810 
809 
808 
807 

20 

21 
22 

23 
24 

825 
960 
8.97  095 
229 
363 

847 
982 

251 

385 

870 
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139 
274 
407 

892 

162 
296 
430 

915 

184 

452 

937 

*°72 

207 
34i 
474 

960 

229 

363 
496 

39 

38 
37 
36 
35 

810 
809 

808 
807 

806 

7 
8 

9 

16.1 
18.4 
20.7 

806 
804 
803 
802 
Boi 

25 
26 

27 
28 

29 

496 
629 
762 
894 
8.98  026 

651 
784 
916 
048 

54i 

674 
806 
938 
070 

563 
696 
828 
960 
092 

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718 
850 
982 
114 

607 
740 

872 

135 

629 

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157 

34 

33 
32 

30 

804 

803 

802 

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800 

i 

2 

3 
4 

22 

2.2 

4-4 
6.6 

8.8 

800 
798 
797 
796 
795 

30 

32 
33 
34 

157 
288 

419 
549 
679 

179 
310 
441 

701 

2OI 
332 
462 
592 
722 

223 

354 
484 
614 
744 

245 
375 
506 
636 
765 

266 
397 
527 
657 

787 

288 
419 

549 
679 
808 

29 

28 

27 
26 

25 

798 

797 

796 

795 
793 

5 
6 

8 
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13.2 

15-4 
17.6 
19.8 

793 
792 
791 
790 

788 

35 
36 

37 
38 
39 

808 

937 
8.99  066 
194 

322 

830 

959 
087 
216 
343 

85I 
980 
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237 
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130 

258 
386 

894 

152 
280 
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173 
301 
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937 
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194 
322 
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24 
23 

22 
21 
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790 

788 
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2 

81 

2.1 

4-2 

787 
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783 
782 

40 

42 
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44 

450 
577 
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598 
725 
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977 

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513 
640 

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534 
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17 
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783 
782 
781 

3 
4 

6 

8 

6.3 

8.4 
10.5 

12.6 

14.7 
16.8 

78i 
780 
778 
111 
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45 
46 

47 
48 
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9.00  082 
207 
332 
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58i 

103 
228 
353 
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601 

123 
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373 
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144 
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165 
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539 
663 

186 

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560 
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207 

332 
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58i 
704 

13 

12 
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10 

780 
778 
111 

7/6 
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18.9 
20 

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773 
772 
771 
769 

50 

52 
53 
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704 
828 

951 
9.01  074 
196 

725 
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971 
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217 

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237 

766 
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135 
257 

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910 

#033 
155 
278 

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930 
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176 
298 

828 
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196 

9 

8 

7 
6 

5 

773 
772 
771 
769 
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2 

3 
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4.0 

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8.0 

1  0.0 
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768 
767 
765 
764 

763 

55 
56 
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58 
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318 
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339 
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359 
480 
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L  Cos    L  Sin 


L  Tan 


5C 


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10" 

20" 

30" 

40" 

50" 

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p 

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0 

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2 

3 
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8-94  195 
340 
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630 

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219 
365 
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725 
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316 
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340 
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630 

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58 
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55 

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2 

3 

25 

2-5 
5.0 

7-5 

6 

7 
8 

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917 
8.95  060 
202 

344 

486 

94i 

083 
226 
368 
509 

964 
107 
249 
391 

533 

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131 
273 
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155 
297 

439 
580 

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178 

320 

462 
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202 

344 

486 
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54 

53 
52 
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50 

4 
6 

8 
9 

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12.5 
15-0 
17-5 

2O.O 
22.5 

10 

ii 

12 
13 
14 

627 

767 
908 
8.96  047 
187 

650 
791 

931 
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2IO 

674 

814 

954 
094 

233 

697 

838 

977 
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256 

721 

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*°°  l 
140 

279 

744 

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163 

302 

767 
908 

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187 
325 

49 
48 
47 
46 
45 

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2 

3 
4 

24 

2.4 
4.8 
7.2 
9.6 

15 

16 

17 

18 

19 

325 
464 
602 

739 

877 

349 
487 
625 
762 
899 

372 
5io 
648 

785 
922 

395 
533 
671 
808 
945 

418 
556 
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831 
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441 

579 
717 

854 
991 

464 
602 
739 

877 
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44 
43 
42 

4i 
40 

6 

8 
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12.0 
144 

16.8 
19.2 

21.6 

20 

21 
22 

23 
24 

8.97013 
150 

285 
421 
556 

036 
172 
308 
443 

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059 
195 
331 
466 
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218 

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488 
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104 
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533 
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39 
38 
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36 
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4.6 
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25 
26 

27 
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8.98  092 
225 

713 
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225 
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34 
33 
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5 
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13.8 
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30 
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32 
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358 

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753 
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380 
512 
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402 

534 
666 

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928 

424 
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446 
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468 
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490 
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29 

28 

27 
26 

25 

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2 

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4 

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22 

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35 
36 
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8.99015 
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275 
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167 
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426 

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058 
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080 

210 
340 

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123 
253 
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512 
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145 

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24 
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174 
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45 
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301 
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322 
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637 

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406 

532 
658 
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427 

553 
679 
805 
930 

14 
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10 

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7 
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12.6 

14.7 

16.8 
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50 
5i 

52 
53 
54 

930 
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179 

303 
427 

95i 
075 
200 

324 

447 

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220 

344 

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117 
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2 

3 
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2.0 
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6.0 
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55 
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550 
673 
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571 
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816 

939 
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162 

4 
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1  6.0 
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60" 

50" 

40" 

30" 

20" 

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P 

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*174°  264°  *354° 


L  Cot 


60 

L  Cos 


L  Sin 


*96°  186°  *276C 


9-99 

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0" 

10" 

20" 

30" 

40" 

50" 

60" 

1 

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761 
760 
759 
757 
756 

0 

i 

2 

3 
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9-01  923 
9.02  043 
163 
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342 
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143 
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59 
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540 
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560 
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54 
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3 
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12.6 

14.7 

16  8 

748 
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745 
744 
742 

10 
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12 
13 
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9-03  109 
226 
342 

458 
574 

128 
245 
361 
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593 

148 

265 
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497 
613 

167 
284 
400 
5i6 

632 

187 
303 
420 

535 
651 

206 
323 
439 
555 
670 

226 
342 
458 
574 
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49 

48 

47 
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747 
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744 

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74i 

9 

18.9 
20 

74i 
740 
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15 

16 

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920 
9.04  034 
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225 

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129 
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730 

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20 

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319 

433 
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727 

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25 
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164 
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182 
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108 
219 
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720 
718 

717 
716 

714 

30 
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32 
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386 
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607 
717 
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515 
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423 
533 
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864 

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460 
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29 

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713 
711 

710 
708 
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35 
36 
37 
38 
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937 
9.06  046 

155 
264 
372 

955 
064 

173 

282 

390 

973 
082 

191 
300 
408 

991 

101 
210 

3I8 
426 

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119 

228 
336 
445 

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137 
246 

354 
463 

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155 
264 
372 
481 

24 
23 

22 
21 
20 

711 
710 
708 
707 

705 

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2 

18 
1.8 
3.6 

705 
704 
702 
701 
699 

40 
4i 

42 
43 
44 

481 

589 
696 
804 
911 

499 
606 
714 
821 
929 

517 
624 
732 

839 
946 

535 
642 

750 
857 
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553 
660 
768 

875 
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678 
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589 
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1  8 

17 
16 

15 

704 
702 
701 
699 
698 

3 
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6 
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5-4 
7-2 
9.0 

10.8 

12.6 

14.4 

698 
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693 
692 

45 
46 

47 

48 

49 

9.07  018 
124 
231 

337 
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035 
142 
248 

354 
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1  60 
266 
372 

478 

071 

177 
284 

39° 
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089 
J95 
301 
407 
513 

106 
213 
319 
425 
530 

124 
231 
337 
442 
548 

14 
13 

12 
II 

10 

696 
695 
693 
692 
690 

9 
j 

16.2 
17 

17 

690 
689 
687 
686 
684 

50 
5i 

52 
53 
54 

548 
653 
758 
863 
968 

566 
671 
776 
881 
985 

583 
688 

793 
898 

*002 

601 
706 
811 
9'5 

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618 

723 

828 

933 
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636 

74i 
846 
950 
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653 
758 
863 
968 

*0"2 

9 

8 

7 
6 

5 

689 
687 
686 
684 
683 

2 

3 

4 

6 

3-4 
5-1 
6.8 

8.5 

10.2 

683 
68  1 
680 
678 
677 

55 
56 
57 
58 
59 

9.08  072 
176 
280 

383 
486 

089 
193 
297 
400 

504 

107 
211 

3U 

418 
521 

124 
228 
331 

435 

538 

141 

245 
349 
452 

555 

159 
262 
366 
469 
572 

I76 
280 

383 
486 

589 

4 
3 

2 
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0 

68  1 
680 
678 
677 
675 

8 
9 

II.9 
13.6 
15-3 

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60' 

40" 

80- 

20" 

10" 

0" 

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9.99 

i 

P 

*173°  263°  *353° 


L  Cos    L  Sin 


rv 

TABLE  OF  THE  LOGARITHMS 


OF   THE 


TRIGONOMETRIC  FUNCTIONS 

FROM  MINUTE  TO  MINUTE. 


0° 


*90°     180°    *270' 


• 

' 

L  Sin 

d 

C  S 

C  T 

LTan 

cd 

L  Cot 

L  Cos 

o 

0 

—00 

—00 

oo 

o.ooooo 

60 

60 

120 
1  80 
240 
300 
360 
420 
480 
540 

I 

2 

3 
4 

6 

i 

6.46  373 
6.76476 
6.94  085 
7.06  579 
7.16270 
7.24  1  88 
7.30  882 
7.36  682 
7-41  797 

30103 
7609 
2494 
9691 
7918 
6694 
5800 
5115 

5-31  443 
5.31  443 
5.31  443 
5.31  443 
5-31  443 
5.31  443 
5-31  443 
5-31  443 
5-31  443 

5-31  443 
5-31  443 
5-31  443 
5.3i  442 
5.3i  442 
5.31  442 
5.31  442 
5.3i  442 
5-31  442 

6.46  373 
6.76  476 
6.94  085 
7.06  579 
7.16  270 
7.24  1  88 
7.30882 
7.36  682 
741  797 

0103 
7609 
2494 
9691 
7918 
6694 
5800 
5"5 
4576 

3.53  627 
3.23  524 

3.05  9J5 
2.93  421 
2.83  730 
2.75812 
2.69  118 
2.63  318 
2.58  203 

o.ooooo 
o.ooooo 

0.00000 
0.00000 

o.ooooo 

0.00000 
0.00000 

o.ooooo 
o.ooooo 

59 
58 
57 
56 
55 
54 
53 
52 
51 

600 

10 

7-46  373 

j.no 

5.31  443 

5-31  442 

746  373 

4139 

2.53627 

0.00000 

50 

660 
720 
780 
840 
900 
960 
1020 
1080 
1140 

II 

12 
13 
14 

15 

16 

17 
18 

19 

7-50512 
7-54  291 
7-57  767 
7.60985 
7.63  982 
7.66  784 

7.69417 
7.71  900 
7-74  248 

3779 
3476 

3218 

2997 
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7-50512 
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2.45  709 
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7.86  166 
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9.99  963 
9.99962 
9.99  962 

9-99  961 
9.99  961 

9.99  960 
9.99  960 
9-99  959 

39 
38 
37 
36 
35 
34 

33 
32 
3i 

9000 

30 

8.63  968 

288 

5-3I456 

5-31415 

8.64  009 

291 

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9.90959 

30 

9O6O 
9120 
9180 
9240 
9300 
9360 
9420 
9480 
9540 

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32 
33 
34 
35 
36 

37 
38 
39 

8.64  256 
8.64  543 
8.64  827 

8.65  no 
8.65  391 
8.65  670 
8.65  947 
8.66  223 
8.66  497 

287 
284 
283 
281 
279 
277 
276 

274 
272 

5-3I456 
5.31457 
5.31457 
5.3M57 
5-31  457 
5-3M57 
5.31458 
5.31458 
5.31458 

5-3I4I5 
5.3i  4H 
5-31  4i4 
5.3i  413 
5.3I4I3 
5.3I4I3 
5-31412 
5-31412 
5-31  412 

8.64  298 
8.64  585 
8.64  870 
8.65  154 
8.65  435 
8-65  715 

8.65  993 
8.66  269 
8.66  543 

287 
285 
284 
28l 
280 
278 
276 
274 

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9.99  958 
9-99  958 
9-99  957 
9-99  956 
9-99  956 
9-99955 
9-99955 
9-99  954 
9-99  954 

29 
28 

27 
26 
25 
24 

23 

22 
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9600 

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8.66  769 

270 

5.3I458 

5-31411 

8.66816 

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9-99  953 

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9660 
9720 
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9900 
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42 
43 
44 
45 
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8.67  039 
8.67  308 
8-67  575 
8.67  841 
8.68  104 
8.68  367 
8.68  627 
8.68  886 
8.69  144 

269 
267 
266 

263 
263 
260 

259 
258 
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5-31  458 
5.3i  459 
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5-31  4io 
5-31  4io 
5-31  409 
5.3i  409 
5-31  408 
5.31  408 
5.31  408 

8.67087 
8.67  356 
8.67  624 
8.67  890 
8.68  154 
8.68417 
8.68  678 
8.68  938 
8.69  196 

271 
269 
268 
266 
264 
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260 
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9-99  952 
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9-99  950 
9-99  949 
9-99  949 
9-99  948 
9-99  948 

19 

18 

17 

16 
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14 
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10440 
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10560 
10620 
10680 
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53 
54 
55 
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57 
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8.69  654 
8.69  907 
8.70159 
8.70  409 
8.70658 
8.70905 
8.71  151 

8.71  395 
8.71  638 

253 
252 
250 

249 
247 
246 

244 
243 

242 

5-31  46o 
5-31  46i 
5-31  46i 
5-31  46i 
5-31  46i 
5-31  461 
5.31  462 
5.31  462 
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5-31  407 
5.31  406 
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5.3i  405 
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5-31  404 
5.3i  404 
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8.69  708 
8.69  962 
8.70214 
8.70  465 
8.70714 
8.70962 
8.71  208 

8.71  453 
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254 
252 
251 
249 
248 
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245 
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9-99  946 
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9.99  944 
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8.72  120 

8.72  181 

1.27  819 

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8.72359 

8.72  597 

239 
238 

8.72420 
8.72  659 

239 
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1.27  580 
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9-99  939 
9.99  938 

58 
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5 
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8.72  834 
8.73  069 
8-73  303 

235 
234 

8.72  896 
8-73  132 
8.73  366 

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236 
234 

1.27  104 
1.26868 
1.26  634 

9-99  938 
9-99  937 
9.99  936 

56 

55 
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28.1 
32.1 
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8-73  535 
8.73  767 

232 

232 

8.73  600 
8.73  832 

234 

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1.26  400 
1.26168 

9-99  936 
9-99  935 

53 
52 

20 

30 
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80.3 
120.5 
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78.0 
117.0 
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8-73  997 

230 

8.74063 

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1.25937 

9-99  934 

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195.0 

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8.74226 

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8.74  292 

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1.25  708 

9-99  934 

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8-74454 

8.74521 

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1.25479 

9.99  933 

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7-6 

7-5 

7-4 

12 
13 

8.74680 
8.74  906 

226 
226 

8.74  748 
8.74  974 

227 

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1.25  252 
1.25  026 

9.99  932 
9.99  932 

48 
47 

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15.0 

18.8 

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14.9 

18.6 

14 

15 

8-75  130 
8.75353 

224 
223 

8.75  199 
8.75423 

225 

224 

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1-24577 

9-99931 
9.99  930 

46 
45 

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22.9 
26.7 
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8.75  645 

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1.24355 

9-99  929 

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8.75  795 

8.75  867 

1-24133 

9.99  929 

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8.76015 

8.76087 

1.23  913 

9.99  928 

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30 

116.0 

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112.5 

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8.76  234 

219 

8.76  306 

219 

1.23  694 

9.99  927 

41 

40 

154.7 

52.7 
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187.5 

148.7 
185.8 

20 

8.76451 

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8.76525 

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1-23475 

9.99  926 

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222 

220 

217 

215 

213 

21 

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8.76  667 
8.76  883 

216 

8.76  742 
8.76958 

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1.23  258 
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9-99  926 
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8.77097 
8.77310 

214 
213 

8-77  173 
8.77387 

215 
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1.22  827 
1.22  613 

9.99  924 
9-99  923 

37 
36 

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5 
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14.7 
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22.  0 

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8.77522 

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8.77  600 

213 

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9-99  923 

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1.22  189 

9.99  922 

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8-77943 

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1.  21  978 

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36.2 

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8.78152 
8.78  360 

209 

208 
208 

8.78232 
8.78441 

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208 

1.  2  1  768 
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9-99  920 

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146.7 

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108.5 
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8.78  568 

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8.78  649 

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9.99  919 

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8.79  061 
8.79  266 

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1.20939 
1.20734 

9.99917 
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28 
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35 
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8.79386 
8.79  588 
8.79  789 

202 
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8.79470 
8-79  673 
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8.80674 

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42 
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8.80  782 
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196 

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8.80872 
8.81  068 
8.81  264 

196 
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I.I9  128 
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1.18  736 

9.99910 
9.99  909 
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12.8 

44 
45 

8.81  367 
8.81  560 

194 

193 

8.81  459 
S.Si  653 

194 

1.18541 
1.18347 

9.99  908 
9-99  9°7 

15 

5 
6 
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16.6 
19.9 
23.2 

16.4 
19.7 
23.0 

16.2 
19-5 

16.1 

19-3 
22.5 

1  6.0 
19.2 
22.4 

46 

8.81  752 

192 

8.  Si  846 

193 

1.18  154 

9.99  906 

14 

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26.5 

26.3 

26.0 

25-7 

25.6 

47 
48 

8.81  944 
8.82  134 

192 
190 

8.82038 
8.82  230 

192 
192 

1.17  962 
1.17  770 

9.99  905 
9.99  904 

13 

12 

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10 

20 

29.8 

29.6 
32.8 
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29.2 
32.5 
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29.0 
32.2 
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28.8 
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8.82  324 

190 

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8.82  420 

190 

1.17  580 

9.99  904 

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8.82  513 

log 

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8.82610 

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I.I7390 

9.99  903 

10 

50 

165.8 

164.2 

162.5 

160.8 

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8.82  701 

8.82  799 

109 

I.I7  201 

9-99  902 

9 

189 

187 

185 

183 

181 

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8.82888 

187 

8.82987 

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I.I70I3 

9.99  901 

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8.83075 

187 

8.83  175 

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I.I6825 

9.99  900 

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9.4 

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9.2 

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8.83  261 
8.83  446 

185 

8.83361 
8.83  547 

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1.16  639 
1-16453 

9.99  899 
9.99  898 

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56 

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8.83  630 
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8.83  732 
8.83916 

184 

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1.16268 
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9.99  898 
9.99  897 

4 
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22.0 
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21.8 

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21.6 

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27.8 

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21.  1 
24.1 
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58 
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8.83  996 
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183 

181 
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8.84  100 
8.84  282 

104 

182 
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1.15900 
1.15  718 

9.99  896 
9.99  895 

2 

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10 

20 

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63.0 

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31.2 

62.3 

93-5 

30.8 
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61.0 
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30.2 
60.3 
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8.84358 

8.84  464 

1.15  536 

9-99  894 

0 

40 

126.0 
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124.7 

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8.84358 

8.84  464 

1.15536 

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8.84  646 

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8.84  897 

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8.85  006 

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9.99  891 

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8.85  075 

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8.85  185 

179 

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9.99  891 

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8.85252 
8.85  429 

177 
177 

8.85  363 
8.85  540 

178 
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9.99  890 
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55 
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24.1 

27.2 
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23.9 
26.8 
29.8 

23.7 

26.7 
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23.6 
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8.85  605 
8.85  780 

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8.85  717 
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9.99  888 
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53 
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118.0 

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8.85955 

175 

8.86  069 

176 

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9.99  886 

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150.8 

149.2 

48.3 

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10 

8.86  128 

173 

8.86  243 

174 

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9.99  885 

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175 

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8.86301 

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8.86417 

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9.99  884 

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8.86474 

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8.86  591 

174 

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8.86  645 

171 

8.86  763 

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9.99  882 

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14.6 

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8.86816 

171 

8.86935 

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9.99  881 

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17.6 

17.4 

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8.86987 

8.87  106 

171 

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9.99  880 

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20.4 
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8.87  156 

169 

8.87277 

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26.2 

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8.87325 

169 

8.87447 

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9.99  879 

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28.8 
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8.87  494 

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8.87616 

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9.99  878 

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8.87661 

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8.87785 

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8.87  829 

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170 

169 

108 

1G7 

21 
22 

8.87  995 
8.88  161 

1  66 

8.88  120 
8.88287 

167 
I67 

.11  880 
•II  713 

9-99  875 
9-99  874 

39 

38 

I 

2 

2.8 

5-7 

8  6 

2.8 

K 

2.8 

5-6 

8  4 

2.8 

5-6 

2.8 

5-6 

23 

8.88326 

165 

8.88453 

1  66 

•ii  547 

9-99  873 

37 

4 

11.4 

"•3 

"•3 

II.  2 

ii.  i 

24 
25 

8.88  490 
8.88654 

164 

8.88618 
8.88  783 

165 

.11  382 
.11  217 

9-99  872 
9-99  871 

36 
35 

7 

14.2 
17.1 

2O.O 

14.2 
17.0 
19.8 

14.1 
16.9 
19.7 

14.0 

16.8 
19.6 

13-9 
16.7 

19.5 

26 

27 

8.88817 
8.88  980 

163 
163 

8.88  948 
8.89  in 

165 
163 

.11  052 
.10  889 

9-99  870 
9.99  869 

34 
33 

8 

9 
10 

22.8 
25.6 

28.5 

22.7 
25-5 
28.3 

22.5 

25-4 
28.2 

22.4 
25.2 
28.0 

22.3 
25.0 
27.8 

28 
29 

8.89  142 
8.89  304 

102 
162 

160 

8.89274 
8.89437 

163 
163 

T6T 

.10726 
.10  563 

9.99  868 
9-99  867 

32 

20 
3° 
40 

57-° 
85.5 
114.0 

5^.7 
85.0 
"3-3 

56.3 
840 
112.7 

56.0 
84.0 

112.0 

55-7 
83.5 
111.3 

30 

8.89  464 

161 

8.89  598 

l62 

.10402 

9.99  866 

30 

5° 

142.5 
166 

141.7 
165 

140.8 
164 

140.0 

163 

139.2 
162 

32 

8.89  625 
8.89  784 

159 

8.89  760 
8.89  920 

1  60 

.10  240 
.10080 

9-99  865 
9.99  864 

29 

28 

i 

2 

3 

2.8 

5-5 
8-3 

2.8 

5-5 

8.2 

2-7 

fcl 

2-7 

i:J 

2.7 

5-4 
8.1 

33 

8.89  943 

I  ^Q 

8.90  080 

1  60 

.09  920 

9.99  863 

27 

4 

ii.  i 

13.8 

II.O 

10.9 
I3-7 

10.9 
13.6 

10.8 

34 

8.90  IO2 

T  tr  ft 

8.90  240 

.09760 

9.99  862 

26 

6 

16.6 

16.5 

16.4 

i6.3 

16.2 

35 

8.90  260 

15° 

8.90  399 

_  _Q 

.09  60  1 

9.99  861 

25 

8 

19.4 

19.2 

19.1 

19.0 

18.9 

21  6 

36 

8.90417 

T  e  7 

8.90  557 

15° 

TCg 

.09  443 

9.99  860 

24 

9 

24.9 

24.8 

24.6 

24.4 

24.3 

37 

38 

8.90  574 
8.90  730 

156 

8.90715 
8.90  872 

157 

.09285 
.09  128 

9-99  859 
9-99  858 

23 

22 

10 

20 

3° 

27.7 

55-3 
83.0 

27-5 
55-0 
82.5 

27-3 
54-7 
82.0 

27.2 
it? 

27.0 
54.0 

8i.c 

39 

8.90  885 

155 

8.91  029 

*5/ 

.08971 

9-99  857 

21 

40 

nc.  7 

IIO.O 

109.3 

\°*i 

108.0 

40 

8.91  040 

155 

8.91  185 

15 

1.08815 

9.99  856 

20 

5° 

I3°-3 
161 

I37-5 
160 

130.7 
159 

I35-° 
158 

157 

41 

8.91  195 

155 

8.91  340 

155 

i.  08  660 

9-99  855 

19 

! 

2.7 

2-7 

2.6 

2.6 

2.6 

42 

8.91  349 

J54 

8.91  495 

155 

1.08  505 

9.99  854 

18 

2 

5-4 

5-3 

11 

5-3 

5-2 

43 

8.91  502 

J53 

8.91  650 

X55 

ICO 

i.  08  350 

9-99  853 

17 

3 

4 

10.7 

10.7 

10.6 

10.5 

10.3 

44 

8.91  655 

8.91  803 

1.08  197 

9-99  852 

16 

5 

31 

13.2 

13-2 
i<;  8 

»3-» 

45 

8.91  807 

152 

8.91  957 

154 

i.  08  043 

9.99851 

15 

18.8 

i«7 

1*3 

18.4 

18.3 

46 

8.91  959 

152 

8.92  no 

153 

1.07  890 

9-99  850 

14 

8 

21.5 

21.2 

21.  1 

20.9 

47 

8.92  no 

151 

8.92  262 

I52 

1.07  738 

9.99  848 

13 

9 
10 

24.2 
26.8 

24.0 
26.7 

23.8 

26.  S 

23-7 
26.3 

T, 

48 

8.92  261 

151 

8.92  414 

152 

1.07  586 

9.99  847 

12 

20 

53-7 

53-3 

53.0 

52.7 

52.3 

49 

8.92411 

15° 

8.92  565 

I51 

1-07435 

9-99  846 

II 

30 

80.5 

80.0 

79-5 

79.0 

78-5 

50 

8.92  561 

8.92  716 

5 

1.07  284 

9-99  845 

10 

134-2 

'33-3 

130.8 

52 

8.92  710 
8.92  859 

149 

T   i  - 

8.92  866 
8.93016 

150 

1.07  134 
1.06  984 

9-99  844 
9-99  843 

I 

i 

156 

2.6 

186 

2.6 

154 

2.6 
e.l 

153 

2.6 

C.I 

2.5 

53 

8.93  007 

140 

8.93  165 

149 
T  ,a 

i.  06  835 

9.99  842 

1 

3 

7-8 

7.8 

7-7 

7-6 

7.6 

54 

55 

8.93  154 
8.93  301 

8.93313 
8.93  462 

149 

i.  06  687 
1.06  538 

9.99  841 
9-99  840 

6 

5 

4 

I 

10.4 
13.0 

10.3 
12.9 
15-5 

10.1 
12.8 

15.4 

IO.2 
12.8 
15-3 

10.  1 

12.7 
15.2 

56 

8.93  448 

I47 
146 

8.93609 

M7 
147 

1.06391 

9-99  839 

4 

7 

.'.'..• 
208 

20.7 

18.0 
20.5 

I7.8 
2O.4 

17.7 

20.3 

57 
58 
59 

8-93  594 
8.93  740 
8.93  8B§ 

146 
MS 
145 

8.93  756 
8.93  903 
8/;4  049 

M7 
146 
146 

1.06244 
1.06097 
1.05951 

9-99  838 
9-99  837 
9-99  836 

3 
2 

I 

9 
10 

20 

••• 

23-4 
26.0 
52.0 
78.0 

23.2 
25.8 
51-7 
77-5 

23.1 

25.7 
51.3 
77.0 

23.0 
25-5 
51.0 
76.5 

22.8 
25-3 
50-7 
76.0 

60 

8.94  030 

8.94  195 

1.05  8o§ 

9-99  834 

0 

104.0 

i  ••.... 

103.3 

i  .•  ,..• 

102.7 

102.0 

XOX.3 

ij&L, 

LCos 

d 

L  Cot 

cd 

L  Tan 

L  Sin 

' 

P 

!< 

•175°    265°    "355° 


67 


*95°  185°  *275° 


' 

L  Sin 

d 

L  Tan 

cd 

LCot 

L  Cos 

P 

P 

0 

8.94  030 

8.94  195 

1.05  805 

9-99  834 

60 

j 

151 

2.5 

149 
2.5 

143 
2.5 

147 
2.4 

146 

2.4 

I 

8.94174 

144 

8-94  34o 

i4i> 

1.05  660 

9-99  833 

59 

2 

3 

7-6 

7-4 

4-9 
7-4 

4-9 
7-4 

4.9 
7-3 

2 

3 

8.94317 
8.94  461 

144 
142 

8.94485 
8.94  630 

H5 
143 

1.05  515 
1.05  370 

9.99  832 
9-99  831 

58 
57 

4 

IO.I 
12.6 

9-9 
12.4 
14.9 

9.9 
12.3 
14.8 

9.8 

12.2 
14.7 

9-7 

12.2 
I4.6 

4 

5 

8.94  603 
8.94  746 

143 

8.94  773 
8.94917 

144 

1.05  227 
1.05  083 

9.99  830 
9.99  829 

56 

55 

7 
8 
9 

17.6 

20.1 
22.6 

17.4 
19.9 
22.4 

17-3 
19.7 

22.2 

17.2 
ig.6 
22.  0 

17.0 

I9-S 
2I.g 

6 

8.94  887 

8.95  060 

1.04  940 

9.99  828 

54 

10 

25.2 

24.8 

24.7 

24'5 

24-3 

7 
8 

8.95  029 
8.95  170 

141 

8.95  202 

8-95  344 

142 

1.04  798 
1.04  656 

9.99  827 
9.99  825 

53 
52 

20 

3° 
40 

50.3 

75-5 
100.7 

49-7 
74-5 
99-3 

49-3 
74.0 

98.7 

49.0 

73-5 
98.0 

48.7 

73-° 
97-3 

9 

8.95  31° 

140 

I  dO 

8.95  486 

IJ.I 

1.04  514 

9.99  824 

51 

50 

125.8 

124.2 

"3-3 

122.5 

121.7 

10 

8-95  450 

8.95  627 

1-04373 

9-99  823 

50 

i 

145 

144 

143 
2.4 

142 

2.4 

141 

2.4 

ii 

8-95  589 

39 

8-95  767 

1.04  233 

9.99  822 

49 

2 

4.8 

4-8 

4.8 

4-7 

4-7 

12 
13 

8.95  728 
8.95  867 

139 

8.95  908 
8.96047 

141 

139 

1.04092 
1.03  95-2 

9.99  821 
9.99  820 

48 
47 

3 
4 

7.2 
9-7 

7.2 
9.6 

7.2 

9-5 

7'1 

9>5 

7.0 

3 

14 
15 

8.96  005 
8.96  143 

138 
138 

8.96  187 
8.96325 

140 

138 

1.03  813 
1-03  675 

9.99  819 
9.99817 

46 
45 

6 

7 
8 

14.5 
16.9 
I9-3 

14.4 

16.8 
19.2 

M-3 
16.7 
19.1 

14.2 
16.6 
18.9 

isia 

16 

17 

18 

19 

8.96  280 
8.96417 

8.96553 
8.96  689 

137 
136 
136 

8.96  464 
8.96  602 
8.96  739 
8.96  877 

138 

137 
138 

TO  A 

1.03  536 
1.03  398 
1.03  261 
1.03  123 

9.99  816 
9.99815 
9.99  814 
9.99813 

44 

43 
42 

9 
10 
20 

40 

21.8 

24.2 
48.3 
72.5 
96.7 

I2O  8 

21.6 

24.0 
48.0 
72.0 
96.0 

21.4 
23.8 
47-7 

95-3 

21.3 

23-7 
47-3 
71.0 

94-7 
118  ^ 

21.2 
23-5 
47.0 
70,5 
94.0 

20 

8.96825 

T-i  e 

8.97  on 

LO2  987 

9.99812 

40 

140 

139 

138 

137 

136 

21 

8.96  960 

8.97  150 

I.O2  850 

9.99  810 

39 

i 

2-3 

2-3 

2.3 

2.3 

2-3 

22 

23 

8-97  095 
8.97  229 

T35 

134 
134 

8.97  285 
8.97421 

T35 
136 

135 

1.02  715 

L02  579 

9.99  809 
9.99  808 

38 
37 

3 

4 

4-7 
7.0 

9-3 

4.6 
7.° 
9-3 

tf 

9.2 

o!8 
9.1 

9.1 

24 

25 

8.97  363 
8.97  496 

133 

8-97  556 
8.97  691 

135 

1.02  444 
1.02  309 

9.99  807 
9.99  806 

36 

35 

7 

14.0 
16.3 

32 

13.8 
16,1 

io!o 

13.6 

26 

8.97  629 

133 

8.97  825 

iid 

1.02175 

9.99  804 

34 

8 
9 

18.7 

2I,O 

18.5 

20.8 

18.4 
20.7 

18.3 

20.6 

18.1 
20.4 

27 

28 
29 

8.97  762 

8.97  894 
8.98  026 

132 

132 
131 

8-97  959 
8.98  092 
8.98  225 

•  en  en  e<" 
•>  en  en  f 

I.O2  041 

i.oi  908 
i.oi  775 

9-99  803 
9.99  802 
9.99  801 

33 
32 

10 
20 

30 
40 

23«3 
46.7 
70.0 

93-3 

23.2 
46.3 
69.5 

92.7 

23.0 
46.0 
69.0 
92.0 

22.8 

45.7 
68.5 

22,7 
45-3 

68.0 
90.7 

30 

8.98157 

131 

8.98358 

132 

i.oi  642 

9.99  800 

30 

5° 

116.7 
135 

115.8 

134 

115.0 
133 

114.2 
132 

113-3 
131 

31 

8.98  288 

8.98  490 

i.oi  510 

9-99  798 

29 

i 

2 

2.2 

4-5 

2.2 
4-S 

2.2 
4.4 

2.2 

4.4 

2.2 

4-4 

32 

8.98  419 

1  J1 

8.98  622 

132 

i.oi  378 

9-99  797 

28 

3 

6.8 

6.7 

6.6 

6.6 

6.6 

33 

8.98  549 

130 
130 

8.98  753 

131 
131 

i.oi  247 

9-99  796 

27 

4 

5 

9.0 

II.  2 

8.9 

II.  2 

8.9 
ii.  i 

8.8 

II.  O 

8-7 
10.9 

34 

8.98  679 

8.98  884 

i.oi  116 

9-99  795 

26 

6 

13-5 

13.4 

13-3 

13.2 

13.1 

35 
36 

8.98  808 
8.98  937 

129 
129 
129 

8.99015 
8-99  145 

130 

i.oo  985 
i.oo  855 

9-99  793 
9-99  792 

25 
24 

7 
8 
9 

15.8 
l8.0 
20.2 

I5.b 
17.9 
20.1 

15.5 
17.7 

2O.O 

15-4 
17.6 
19.8 

17-5 
19.6 

37 
38 

8.99  066 
8.99  194 

128 

128 

8.99275 
8.99  405 

130 

LOO  72§ 

LOO  595 

9.99  791 
9-99  790 

23 

22 

10 

20 

3° 

22.5 

45,o 
67.5 

22.3 
44-7 
67.0 

22.2 

44-3 
66.  S 

22.O 

44-o 

66.0 

21.8 

43-7 
65-5 

39 

8.99  322 

128 

8.99  534 

129 

128 

i  .00  466 

9-99  788 

21 

40 

90.0 

89.3 

88.7 

88.0 

87.3 

40 

8-99  450 

127 

8/10  662 

120 

1.00338 

9-99  787 

20 

5° 

112.5 
130 

III.7 

129 

128 

127 

126 

41 

8-99  577 

8-99  791 

i.oo  209 

9-99  786 

19 

i 

2.2 

2.2 

2.1 

2.1 

2.1 

42 
43 

8.99  704 
8.99  830 

127 
126 
126 

8.99  919 
9.00  046 

127 
128 

LOO  OS  I 

0.99  954 

9-99  785 
9.99  783 

18 
17 

2 

4 

tl 

8.7 

3 

8.6 

it 

8.5 

4.2 
6.4 

8.S 

4.2 

6-3 
8.4 

44 

8-99  956 

T  r>A 

9.00  1  74 

0.99  826 

9.99  782 

16 

5 

10.8 

10.8 

10.7 

10.6 

10.5 

45 
46 

9.00  082 
9.00  207 

125 

9.00  301 
9.00427 

127 
126 
126 

0.99  699 
0.99  573 

9.99  781 
9.99  780 

15 
14 

6 
7 
8 

13.0 
17-3 

12.9 
15-0 
17.2 

12.8 

14.9 
17.1 

12.7 
14.8 
16.9 

12.6 

14.7 
16.8 

47 

9.00  332 

TO  A 

9-00  553 

126 

0.99  447 

9-99  778 

13 

9 
10 

J9.5 
21.7 

19.4 
21.5 

19.2 
21.3 

19.0 

21.2 

18.9 

21.  0 

4° 

9.00456 

9.00  679 

0.99  321 

9-99  777 

12 

20 

43-3 

43-0 

42.7 

42.3 

42.0 

49 

9.00  581 

123 

9.00  805 

•roc 

0.99  195 

9-99  776 

II 

30 

05.0 

64.5 

64.0 

63.5 

63.0 

50 

9.00  704 

I2d 

9.00  930 

TOC 

0.99  070 

9-99  775 

10 

40 
50 

108.3 

I07-5 

»5-3 
106.7 

10=5.8 

105.0 

51 

9.00  828 

9.01  055 

0.98  945 

9-99  773 

9 

125 

124 

123 

122 

121 

52 

9.00  951 

123 

9.01  179 

124 

0.98  821 

9-99  772 

8 

I 

2.1 

2.1 

2.O 

2.O 

2.O 

53 

9.01  074 

123 
122 

9-01  303 

124 
124 

0.98  697 

9.99  77i 

7 

3 

4.2 
6.2 

4.1 

6.2 

4-1 
6.2 

4-1 

6.1 

4.0 

6.0 

54 

55 

9.01  196 
9.01  318 

122 

9.01  427 
9-01  550 

123 

0.98  573 
0.98  450 

9-99  769 
9-99  768 

6 

5 

4 

8-3 
10.4 
12.5 

8.3 
10.3 
12.4 

8.2 
10.2 
12.3 

8.1- 

IO.2 
12.2 

8.1 

IO.I 
12.  1 

56 

9.01  440 

9.01  673 

123 

0.98  327 

9-99  767 

4 

7 

14.6 

14.4 

I4.2 

I4.I 

57 

9.01  561 

9.01  796 

123 

0.98  204 

9-99  765 

3 

8 
9 

16.7 
18.8 

10.5 
T8.6 

ld.4 

l8.4 

I0.3 

16.1 
18.2 

58 

9.01  682 

9.01  918 

0.98  082 

9-99  7^4 

2 

10 

20.8 

20.7 

20.5 

20.3 

20.  2 

59 

9.01  80^ 

1  2O 

9.02  040 

122 

0.97  960 

9-99  763 

I 

20 

3° 

41.7 

62.5 

4r-3 
62.0 

41.0 

40.7 

61.0 

40-3 
60.5 

60 

9.01  923 

9.02  162 

0.97  838 

9.99  761 

0 

40 

5° 

83-3 
104.2 

82.7 

82.0 

81.3 

TOT.  7 

80.7 
T00.8 

L  Cos 

d 

L  Cot 

cd 

L  Tan 

L  Sin 

' 

P 

P 

*174°  264°  *354< 


68 


*96°  186°  *276° 


• 

LSin 

d 

L  Tan 

cd 

L  Cot 

LCos 

p  p 

0 

9.01  923 

9.02  162 

0.97  838 

9.99  761 

60 

I 

9.02  043 

1  20 

9.02  283 

121 

0.97717 

9.99  760 

59 

121 

120 

119 

118 

2 

9.02  163 

1  20 

9.02  404 

121 

0.97  596 

9-99  759 

58 

i 

2.0 

2.O 

2.0 

2.O 

3 

9.02  283 

I  IQ 

9.02  525 

1  2O 

0-97475 

9-99  757 

57 

2 

4.0 

4.0 

4.0 

3.9 

4 

9.02  402 

TTc 

9.02  645 

0-97  355 

9-99  756 

56 

3 

6.0 

6.0 

6.0 

5-9 

5 

9.02  520 

9.02  766 

0.97  234 

9-99  755 

55 

4 

8.1 

8.0 

7-9 

7-9 

6 

9.02  639 

119 

118 

9.02  885 

ny 
I2O 

0.97115 

9-99  753 

54 

5 

IO.I 

10.0 

9-9 

Q.b 

7 

8 

9 
10 

9.02  757 
9.02  874 
9.02  992 

117 
118 
117 

9-03  005 
9.03  124 
9.03  242 

118 
H9 
118 

0.96  995 
0.96  876 
0.96  758 

9-99  752 
9-99  75i 
9.99  749 

53 
52 
51  ' 
50 

6 

8 
9 

IO 

12.  1 
I4.I 

16.1 

18.2 

2O  2 

I2.O 
14.0 

16.0 
1  8.0 

11.9 
13-9 
15.9 

17.8 
19  8 

ii.  8 
13.8 

15-7 
17.7 

IQ  7 

9-03  109 

9-03  361 

0.96  639 

9.99  748 

n 

9.03  226 

116 

9-03  479 

118 

0.96  521 

9-99  747 

49 

20 

40.3 

40.0 

39-7 

39'3 

12 
13 

9-°3  342 
9-03  458 

116 

TT6 

9-03  597 
9.03  714 

117 
118 

0.96  403 
0.96  286 

9-99  745 
9-99  744 

48 
47 

30 
40 

60.5 
80.7 

60.0 
80.0 

59-5 

79-3 

59-° 

78.7 

14 

9-°3  574 

116 

9.03  832 

0.96  168 

9.99  742 

46 

50 

100.8 

IOO.O 

QQ.2 

98.3 

15 

9-03  690 

TTC 

9.03  948 

0.96052 

9.99  741 

45 

16 

9.03  805 

9.04  065 

11  7 
116 

0-95  935 

9.99  740 

44 

117 

116 

115 

114 

17 

9.03  920 

114 

9.04  181 

116 

0.95  819 

9-99  738 

43 

i 

2.O 

1.9 

1-9 

18 

9.04  034 

115 

9.04  297 

116 

0-95  703 

9-99  737 

42 

2 

3.9 

3-9 

3.8 

3-8 

19 

9.04  149 

9.04413 

0-95  587 

9.99  736 

41 

3 

5.8 

5.8 

5.8 

5-7 

20 

9.04  262 

I  Id 

9.04  528 

ne. 

0.95  472 

.  9-99  734 

40 

4 

7-8 

7-7 

7-7 

76 

21 

9.04  376 

9.04  643 

0-95357 

9-99  733 

39 

5 

9-8 

9-7 

9.6 

9-5 

22 

9.04  490 

9.04  758 

1*5 

0.95  242 

9-99  731 

38 

6 

ii-7 

11.6 

ii.  5 

11.4 

23 

9.04  603 

9.04  873 

H5 

0.95  127 

9-99  730 

37 

7 

13-6 

13-5 

13-4 

13.3 

24 

9.04715 

9.04987 

114 

0.95  013 

9.99  728 

36 

8 

15-6 

15-5 

15-3 

15.2 

25 

9.04  828 

9.05  101 

H4 

0.94  899 

9-99  727 

35 

9 

17.6 

17.4 

17.2 

17.1 

26 

27 
28 

29 

9.04  940 

9.05  052 
9.05  164 
9-05  275 

112 
112 
III 
III 

9-05  214 
9.05  328 
9.05  441 
9-°5  553 

114 
112 

0.94  786 
0.94672 

0-94  559 
0.94  447 

9-99  726 
9.99  724 
9-99  723 
9-99  721 

34 

33 
32 
31 

IO 
20 
30 
40 
50 

iy-b 

39-° 
58.5 
78.0 

97-5 

19-3 
38.7 
58.0 

77-3 
96.7 

19.2 

38.3 
57-5 
76.7 
95-8 

19.0 

38-0- 
57-0 
76.0 
95-o 

30 

9.05  386 

III 

9.05  666 

112 

0-94  334 

9-99  720 

30 

110 

1  in 

31 

9-05  497 

I  IO 

9.05  778 

0.94  222 

9.99  718 

29 

I 

110 

I.Q 

1J.SS 

I.Q 

1.8 

11U 

1.8 

32 

9-05  607 

9.05  890 

0.94  1  10 

9.99  717 

28 

2 

3  8 

a  7 

3-7 

3-7 

33 

9.05  717 

no 

9.06  002 

III 

0.93  998 

9-99  7i6 

27 

3 

5.6 

5.6 

5-6 

5-5 

34 

9.05  827 

9.06  113 

0.93  887 

9-99  714 

26 

4 

7-5 

7-5 

7-4 

7-3 

35 

9-05  937 

TOO 

9.06  224 

0.93  776 

9.99  713 

25 

5 

9-4 

9-3 

Q.2 

9-2 

36 

9.06  046 

IO9 

9-o6  335 

no 

0.93  665 

9.99  711 

24 

6 

II-3 

II.  2 

II.  I 

II.O 

37 

9-06  155 

9.06  445 

o.93  555 

9.99  710 

23 

7 

13-2 

I3-I 

13-0 

12.8 

38 

9.06  264 

9-06  556 

0.93  444 

9-99  708 

22 

8 

I5-I 

14.9 

14.8 

14.7 

39 
40 

42 
43 

9.06  372 

I09 

108 

107 

108 
107 

9.06  666 

109 

no 
109 
109 
108 

0-93  334 

9-99  707 

21 
20 

19 
18 

17 

9 

10 
20 
30 
40 
5O 

17.0 
18.8 
37-7 
56.5 
75-3 
94.2 

16.8 
18.7 
37-3 
56.0 

74-7 

16.6 
18.5 
37-0 
55-5 
74.0 

Q2.C, 

16.5 
18.3 
36.7 
55.0 

73-3 

QI.7 

9.06481 

9.06  775 

0.93  225 

9-99  705 

9-06  589 
9.06  696 
9.06  804 

9.06  885 
9.06  994 
9-07  103 

0.93  115 
0.93  006 
0.92  897 

9-99  704 
9-99  702 
9-99  70i 

44 

9.06911 

9.07211 

0.92  789 

9.99  699 

16 

45 
46 

9.07  018 
9.07  124 

107 
106 
107 

9-07  320 
9.07  428 

109 
108 
108 

0.92  680 
0.92  572 

9.99  698 
9.99  696 

15 
14 

109 

T  8 

108 

i  8 

107 

i  8 

106 

i  8 

47 
48 
49 

9.07  231 
9-07  337 
9.07  442 

1  06 
105 
1  06 

9-07  536 
9.07  643 
9.07  751 

107 
108 

IO7 

0.92  464 
0.92  357 
0.92  249 

9-99  695 
9.99  693 
9-99  692 

13 

12 
II 

2 

3 

3.6 

5-4 
7  3 

3-6 

5-4 

7.2 

3-6 

5-4 
7.1 

3-5 
5.3 
7.1 

50 

9.07  548 

TOC. 

9.07  858 

1  06 

0.92  142 

9.99  690 

10 

5 

9.1 

9.0 

8.9 

8.8 

51 

9.07  653 

9.07  964 

0.92  036 

9-99  689 

9 

6 

10.8 

10.7 

10.6 

52 

9.07  758 

5 

9.08  071 

107 

0.91  929 

9-99  687 

8 

7 

12.7 

12.6 

12.5 

12.4 

53 

9.07  863 

105 

9.08  177 

1  06 

0.91  823 

9.99  686 

7 

8 

14-5 

14.4 

14-3 

14.1 

54 
55 
56 

9.07  968 
9.08  072 
9.08  176 

104 
104 

9.08  283 
9.08  389 
9.08  49! 

106 
106 

0.91  717 
0.91  6n 
0.91  505 

9.99  684 
9.99  683 
9.99  68  1 

6 

5 
4 

9 

IO 
20 

16.4 

18.2 

36.3 

16.2 
1  8.0 
36.0 

16.0 
17.8 

35-7 

15-9 
17-7 
35-3 

57 
58 
59 

9.08  280 
9-o8  383 
9.08  486 

104 

103 
103 

103 

',.'--  1,1,0 

9.08  705 

9.08  8  10 

105 

105 
105 
104 

0.91  400 
0.91  295 
0.91  i'>'> 

9.99  680 
9.99  678 
9.99  677 

3 
2 

I 

30 
40 
50 

54.5 
72.7 

90.8 

54-0 
72.0 
90.0 

53.5 
7L3 
89.2 

53-0 
70.7 
88.3 

60 

9.08  589 

<;.""  '(i.J 

9-99  675 

0 

LCos 

d 

LCot 

cd 

L  Tan 

L  Sin 

' 

P   P 

•173"  203°  »353° 


*97°  187°  *277C 


L  Sin 

d 

L  Tan 

cd 

L  Cot 

L  Cos 

P  P 

0 

9.08  589 

g.o8  gi4 

10 

o.gi  086 

9-99  675 

60 

10K 

104 

103 

102 

I 

g.o8  6g2 

g.og  oig 

o.gogSi 

g.gg  674 

59 

I 

1.8 

2 

9-o8  795 

1UJ 

g.og  123 

104 

o.go  877 

g.gg  672 

58 

2 

3-5 

3-5 

34 

34 

3 

g.o8  8g7 

IO2 

g.og  227 

104 

o.go  773 

g.gg  670 

57 

3 

5-2 

5.2 

5-2 

5.i 

i  4 

g.o8  ggg 

9.09  330 

103 

o.go  670 

g.gg  669 

56 

4 

7-0 

6.9 

6.9 

6.8 

5 

g.og  101 

g.og  434 

104 

o.go  566 

g.gg  667 

55 

5 

8.8 

8-7 

8.6 

8.5 

6 

g.og  202 

IO2 

9-09  537 

103 

o.go  463 

g.gg  666 

54 

6 

10.5 

10.4 

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IO.2 

7 

g.og  304 

g.og  640 

* 

o.go  360 

g.gg  664 

53 

7 

12.2 

12.  1 

12.0 

n.g 

8 

g.og  405 

g.og  742 

IO2 

o.go  25  8 

g.gg  663 

52 

8 

14.0 

13-9 

13.7 

13.6 

9 

g.og  506 

IOO 

g.og  845 

IO3 

o.go  1  55 

g.gg  661 

51 

9 

15-8 

15.6 

154 

15.3 

10 

g.og  606 

g.og  g47 

o.go  053 

g.gg  659 

50 

IO 

17.5 

17-3 

17.2 

17.0 

ii 

g.og  707 

g.ioo4g 

o.Sggsi 

g.gg  658 

49 

20 

35.0 

34-7 

34.3 

34.0 

12 

g.og  807 

g.io  150 

o.8g  850 

g.gg  656 

48 

30 

52.5 

52.0 

51.5 

51-0 

13 

g.og  go7 

g.io  252 

o.8g  748 

9-99  655 

47 

40 

70.0 

69.3 

68.7 

68.0 

14 

g.io  006 

99 

9.10353 

IOI 

o.8g  647 

g.gg  653 

46 

50 

87-5 

86.7 

85.8 

85.0 

15 

1  6 

g.io  106 
g.io  205 

99 

QQ 

g.io  454 
9-I0555 

IOI 
IOI 

0.89  546 
0.89  445 

g.gg  651 
g.gg  650 

45 
44 

j 

101 

I  7 

100 

I  7 

99 

i  6 

98 
i  6 

17 

18 

19 

g.io  304 
g.io  402 
g.io  501 

98 

99 
08 

g.io  656 
g.io  756 
g.io  856 

IOO 
IOO 

0.89  344 
0.89  244 
0.89  144 

g.gg  648 
g.gg  647 
g.gg  645 

43 

42 

2 

3 
4 

34 
5-0 
6.7 

3-3 
5-0 
6.7 

3.3 
5-0 
6.6 

3.3 
4.9 

20 

g.io  sgg 

go 

08 

g.iog56 

0.89  044 

g.gg  643 

40 

84 

8  3 

8  2 

8  2 

21 

g.io  6g7 

go 

nR 

g.n  056 

0.88  944 

g.gg  642 

39 

6 

IO.I 

IO.O 

Q.Q 

9.8 

22 

g.io  7gs 

go 

9-11  J55 

99 

0.88  845 

g.gg  640 

38 

7 

n.  8 

11.7 

II.6 

II.  4 

23 

g.io  8g3 

97 

g.n  254 

99 

QQ 

0.88  746 

g.gg  638 

37 

8 

13.5 

13-3 

13.2 

24 

g.ioggo 

9-"  353 

0.88  647 

g.gg  637 

36 

9 

15.2 

15.0 

14.8 

14.7 

25 

g.n  087 

97 

9-11452 

99 

0.88  548 

9-99  635 

35 

IO 

16.8 

16.7 

16.5 

26 

g.n  184 

97 
97 

9.11  551 

99 
08 

0.88  44g 

9-99  633 

34 

20 

33-7 

33-3 

33-0 

32.7 

27 

g.n  281 

g.n  649 

nR 

0.88351 

g.gg  632 

33 

30 

50.5 

50.0 

49-5 

49-0 

28 

9-II377 

96 

g.n  747 

9° 

nR 

0.88  253 

g.gg  630 

32 

40 

67,3 

66.7 

66.0 

65.3 

29 

g.n  474 

97 

g.n  845 

98 

0.88  155 

g.gg  62g 

31 

50 

84.2 

83.3 

82.5 

30 

g.n  570 

go 

nfi 

g.n  943 

9 

0.88057 

g.gg  627 

30 

31 

g.n  666 

go 

g.  1  2  040 

97 

0.67  g6o 

g.gg  625 

2Q 

97 

ye 

95 

94 

32 

9.11  761 

95 

g.12  138 

9° 

0.87  862 

g.gg  624 

28 

i 

1.0 

1.6 

1.6 

1.6 

33 

g.n  857 

96 

g.i2  235 

97 

0.87  765 

g.gg  622 

27 

2 

3-2 

3-2 

3-2 

3.1 

34 
35 

9.H952 
9.12  047 

95 
95 

g.i2  428 

97 
96 

0.87  668 
0.87  572 

g.gg  620 
g.gg  618 

26 

25 

3 
4 

4.8 

6.5 

4.8 

6.4 

4.8 
6.3 

4-7 
6-3 

36 

9.12  142 

95 

g.12  525 

97 

0.87475 

g.gg  617 

24 

5 

8.1 

8.0 

7-9 

7.8 

37 
38 
39 
40 

42 
43 

9.12  236 

9-12331 
9.12425 

94 
95 
94 
94 
93 
94 
93 
93 

g.i2  621 
g.12  813 

96 
96 
96 
96 
95 
95 
95 
95 

0.87379 
0.87283 
0.87  187 

9-99615 
g.gg  613 
g.gg  612 

23 

22 
21 
20 

18 

17 

b 

7 
8 

9 

IO 
20 
30 
40 

9«7 

12.9 
14.6 
16.2 
32.3 
48.5 
64.7 

g.b 

II.  2 

12.8 

14.4 

1  6.0 
32.0 
48.0 
64.0 

9-5 
ii.  i 
12.7 
14.2 

15-8 
31.7 
47-5 
63.3 

94 

II.O 

12.5 
14.1 

15.7 
31-3 
47.0 
62.7 

g.12  5ig 

g.i2  gog 

0.87  ogi 

g.gg  610 

g.12  612 
9.12  706 
9.12  7gg 

9.13  004 
g.i3  ogg 
9.13  194 

0.86  gg6 
0.86  goi 
0.86  806 

g.gg  608 
g.gg  607 
g.gg  605 

44 

g.12  892 

g.i3  289 

0.86  711 

g.gg  603 

16 

5° 

80.8 

80.0 

7Q.2 

78.1 

45 

g.12  g85 

93 

9.J3384 

95 

0.86616 

g.gg  601 

15 

46 

9-13078 

93 

9.13478 

94 

0.86  522 

g.gg  600 

14 

93 

92 

91 

90 

47 

9.13  171 

93 

9-1-3  573 

95 

0.86427 

g.gg  5g8 

13 

i 

1.6 

1.5 

1-5 

1.5 

48 

9.13  263 

92 

9.13667 

94 

0.86333 

g.gg  5g6 

12 

2 

3.1 

3.1 

3-0 

3-o 

49 

9-I3355 

92 

g.i3  761 

94 

0.86  23g 

9-99  595 

II 

3 

4.6 

4.6 

4.6 

4-5 

50 

9-  T  3  447 

92 

9-13  854 

93 

0.86  146 

9-99  593 

10 

4 

6.2 

6.1 

6.1 

6.0 

51 

9-1-3  539 

92 

g.i3  g48 

94 

0.86052 

9-99  591 

g 

5 

7-8 

7-7 

7-6 

7-5 

52 

9-13  630 

91 

9.14041 

93 

0.85  gsg 

g.gg  s8g 

8 

6 

9-3 

g.2 

9.1 

g.o 

53 

g.i3  722 

92 

9-1-4  134 

93 

0.85  866 

g.gg  588 

7 

7 

10.8 

10.7 

10.6 

10.5 

54 

9.13  813 

91 

9.14227 

93 

0.85  773 

g.gg  586 

6 

8 

12.4 

12.3 

12.  1 

I2.O 

55 

9.13  904 

91 

9.14  320 

93 

0.85  680 

g.gg  584 

5 

9 

14.0 

13-8 

13.6 

13-5 

56 

9-1-3  994 

go 

9.14412 

92 

0.85  588 

g.gg  582 

4 

10 

15-5 

15-3 

15.2 

15.0 

57 
58 
59 

g.i4o8s 
g.i4266 

91 
go 

9.14  504 

9-J4597 
9.14688 

92 
93 
91 

0.85  4g6 
0.85  403 
0.85  312 

g.gg  581 
9-99  579 

3 

2 

I 

20 

30 
40 

31-0 
46.5 
62.0 

30-7 
46.0 
61.3 

30.3 

45.5 
60.7 

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30.0 

45-0 
60.0 

60 

9-I4356 

go 

9.14  780 

92 

0.85  220 

9-99  575 

0 

5° 

77-5 

75-0 

75-° 

LCos 

d 

L  Cot 

cd 

L  Tan 

L  !Sin 

' 

P  P 

262°  *352C 


*96°  188C  *2?8° 


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L  Sin 

d 

L  Tan 

cd 

L  Cot 

LCos 

p  p 

0 

9-I4356 

QO 

9.14780 

02 

0.85  220 

9-99  575 

60 

92 

91 

90 

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9-14445 

°y 

9.14872 

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0.85  128 

9-99  574 

59 

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i.5 

i.5 

1-5 

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9.14535 

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Rn 

9.14963 

91 

0.85  037 

9-99  572 

58 

2 

3.1 

3-0 

3-0 

3 

9.14  624 

°9 

9.15054 

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0.84  946 

9-99  570 

57 

3 

4.6 

4.6 

4-5 

4 

9.14  714 

9° 

Rn 

9^5  145 

91 

0.84855 

9-99  568 

56 

4 

6.1 

6.1 

6.0 

5 

9.14  803 

89 
QO 

9.15  236 

91 

0.84  764 

9-99  566 

55 

5 

7-7 

7.6 

7-5 

6 

9.14  891 

So 

9.I5327 

91 

0.84  673 

9-99  56! 

54 

9.2 

9.1 

9.0 

7 

9.14980 

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Rn 

9.15417 

9U 

0.84  583 

9-99  563 

53 

7 

10.7 

10.6 

10.5 

8 

9.15069 

89 

OQ 

9.15  508 

91 

0.84  492 

9.99  56i 

52 

8 

12.3 

12.  1 

12.0 

9 

9-15  157 

88 

9-!5  598 

9° 
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0.84  4O2 

9-99  559 

5i 

9 

13-8 

13.6 

13-5 

10 

9-15245 

88 

9.15688 

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80 

0.84312 

9-99  557 

50 

10 

15-3 

15.2 

15.0 

ii 

9.15333 

QQ 

9-15  777 

0.84  223 

9-99  556 

49 

20 

30.7 

30.3 

30.0 

12 

13 

9.15421 
9.15508 

87 

QQ 

9.15  867 
9.15956 

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89 

0.84  133 
0.84  044 

9-99  554 
9-99  552 

48 
47 

30 
40 

46.0 
61.3 

45.5 

60.7 

45.0 
60.0 

14 

9.15  596 

87 

9.16046 

9° 

Rn 

0.83  954 

9-99  550 

46 

50 

76.7 

75-8 

75-0 

15 

9.15683 

°/ 

Q- 

9-16135 

°9 
Rn 

0.83  865 

9-99  548 

45 

^9 

88 

87 

16 

9.15  770 

07 

87 

9.16  224 

»9 

88 

0.83  776 

9-99  546 

44 

i 

1-5 

i.5 

i-4 

17 

9«!5  857 

Bt. 

9.16312 

Rn 

0.83  688 

9-99  545 

43 

2 

3-0 

2.9 

2-9 

18 

9-  i  5  944 

°7 

QA 

9.16401 

89 

RR 

0.83  599 

9-99  543 

42 

3 

44 

4-4 

4-4 

19 

9.16030 

86 

9.16489 

88 

0.83  511 

9-99  54i 

4i 

4 

5-9 

5.9 

5-8 

20 

9.16  116 

87 

9.16577 

88 

0.83  423 

9-99  539 

40 

5 

7-4 

7-3 

7-2 

21 

9.16203 

CA 

9.16665 

RR 

0.83335 

9-99  537 

39 

6 

8.9 

8.8 

8-7 

22 

9.16289 

Qc 

9-I6753 

RR 

0.83  247 

9-99  535 

38 

7 

10.4 

10.3 

10.2 

23 

9.16374 

°D 

86 

9.16  841 

«7 

0.83  159 

9-99  533 

37 

8 

11.9 

ii.  7 

11.6 

24 

9.16460 

Re 

9.16  928 

RR 

0.83  072 

9-99  532 

36 

9 

13-4 

13-2 

13-0 

25 

9-16545 

°5 

CA 

9.17  016 

Q7 

0.82  984 

9-99  530 

35 

10 

14.8 

14.7 

14-5 

26 

9.16631 

85 

9.17103 

°7 

87 

0.82  897 

9.99  528 

34 

20 

29-7 

29.3 

29.0 

27 

9.16  716 

Re 

9.17190 

Q7 

0.82  810 

9-99  526 

33 

30 

44-5 

44.0 

43-5 

28 

9.16801 

°5 
Re 

9.17277 

°7 

Of. 

0.82  723 

9-99  524 

32 

40 

59-3 

58.7 

58.0 

29 

9.16886 

°5 
Q, 

9-I7363 

RT 

0.82  637 

9-99  522 

3i 

50 

74.2 

73-3 

72.5 

30 

9.16  970 

°4 

Br 

9.I7450 

86 

0.82  550 

9.99  520 

30 

86 

85 

84 

3i 

9-J7055 

°D 

Ri 

9-I7536 

Rn 

0.82  464 

9-99  5i8 

29 

I      T  1 

I  4. 

I  4. 

32 
33 

9.17139 
9-17223 

04 

84 

84 

9.17622 
9.17708 

86 
86 

0.82  378 
0.82  292 

9-995I7 
9-995I5 

28 
27 

2 

3 

2-9 
4-3 

2.8 
4-2 

2.8 

4-2 

34 
35 
36 

9.17307 
9.17391 
9.17474 

84 
83 
84 

9.17  794 
9.17  880 
9.17965 

86 

85 
86 

0.82  206 
0.82  1  20 
0.82035 

9-99  513 
9.99511 

9-99  509 

26 

25 
24 

4 
6 

5-7 
7-2 
8.6 

5.7 

7-i 
8.5 

5-6 
7-0 
8.4 

37 

38 

9.I7558 
9.17641 

83 
8-* 

9.18051 
9.18  136 

85 
8c. 

0.81  949 
0.8  1  864 

9-99  507 
9-99  505 

23 

22 

7 
8 

IO.O 

11.5 

9.9 
ii-3 

9.8 

II.  2 

39 

9.17724 

8~ 

9.18  221 

Re 

0.81  779 

9-99  503 

21 

g 

12.9 

12.8 

12.6 

40 

4i 
42 

43 

9.17807 

83 

83 
82 
82 

9.18  306 

•>  \n  tf  u->  -i 

>  00  00  CO  CO 

0.81  694 

9-99  50i 

20 
19 

18 
17 

10 
20 
30 
40 

14-3 
28.7 
43-o 

57-3 

14.2 

28.3 
42.5 
56.7 

14.0 
28.0 
42.O 
56.0 

9.17890 

9-17973 
9-18055 

9.18391 
9.18475 
9.18  560 

0.81  609 
0.81  525 
o.Si  440 

9-99  499 
9-99  497 
9-99  495 

44 

9.18137 

Ra 

9.18  644 

R  A 

0.81  356 

9-99  494 

16 

50 

71-7 

70.8 

7O.O 

45 

9.18  220 

°J 

On 

9.18  728 

°4 

R  i 

o.Si  272 

9-99  492 

15 

46 

9.18  302 

RT 

9.18  812 

°4 
c  . 

0.81  188 

9-99  49° 

14 

83 

82 

81 

M 
48 
49 

9-18383 
9.18465 
9.18  547 

82 
82 

RT 

9.18  896 

9.18979 
9.19063 

°4 
83 

84 

ST 

0.81  104 
0.81  021 
0.80937 

9-99  488 
9.99  486 
9-99  484 

13 

12 
II 

i 

2 

3 

1.4 

2.8 

4.2 

1-4 
2.7 
4.1 

1.4 

2.7 
4-0 

50 

5i 
52 

9.18628 

81 
81 

RT 

9.19  146 

°J 

83 
83 

Ri 

0.80  854 

0.09  482 

10 

9 

8 

4 
6 

5.5 
6.9 

8.3 

5-5 
6.8 

8.2 

5-4 
6.8 
8.1 

9.18  709 
9.18790 

9.19  229 
9.19312 

0.80  771 
0.80688 

9-99  48o 

9-094/8 

53 

9.18  871 

RT 

9-19395 

83 

Ri 

0.80  605 

9-99  47t> 

7 

7 

9-7 

9.6 

91 

54 
55 

9.18952 
9.I9033 

Si 

Rn 

9.19478 
9.19561 

°J 
83 

&2 

0.80  522 
0.80  439 

9-99  474 

6 

5 

8 
9 

ii.  i 

12.4 

10.9 
12.3 

10.8 

12.2 

56 

9.19113 

Rn 

9.19643 

82 

0.80357 

9-99  470 

4 

10 

13-8 

13.7 

13-5 

57 
58 
59 
60 

9.19193 
9.19273 
9-19353 

Bo 

80 
80 

9-r9  725 
9.19807 
9.19889 
g.io.  971 

82 
82 
82 

0.80275 
0.80  193 
0.80  1  1  1 

9.99  468 
9.99  464 

\l 

i 
0 

20 
30 
40 
50 

27.7 
4i-5 
55-3 
69.2 

27-3 
41.0 

54-7 
68.3 

27.0 
40.5 
54.0 
67.5 

M  i  '  i  i  ;  ; 

0.80  029 

•  .  •>  !<•-' 

L  Cos 

d 

L  Cot   <•  «1 

L  Tan 

1,  Sin 

P  P 

•171°  261°  "3510 


71 


*99°  189°  *279C 


' 

L  Sin 

d 

L  Tan 

cd 

L  Cot 

L  Cos 

p  p 

0 

9.19433 

9.19971 

0.80  029 

9.99  462 

GO 

I 

2 

9'*9  Si3 
9.19  592 

79 

9.20053 
9.20  134 

82 

81 

0-79  947 
0.79  866 

9.99  460 
9.99458 

59 

58 

80 

79 

78 

77 

3 

9.19  672 

79 

9.20216 

82 
81 

0-79  784 

9.99  456 

57 

2 

2.7 

2.6 

2.6 

2.6 

4 

5 

9.19  830 

79 

9.20  297 
9.20378 

81 

0.79  703 
0.79  622 

9-99  454 
9-99452 

56 

55 

3 
4 

4.0 

5-3 

4.0 

5-3 

3-9 

5-2 

3.8 
5.1 

6 

9.19  909 

79 

7O 

9.20459 

QT 

0.79  541 

9-99  450 

54 

6-7 

6.6 

6.5 

6.4 

7 

9.19988 

/y 

9-20  540 

0.79460 

9-99  448 

53 

6 

8.0 

7-9 

7.8 

7-7 

8 

9.20  067 

79 

7ft 

9.20  621 

81 

0-79379 

9-99  446 

52 

7 

9-3 

9.2 

9.0 

9 

9.20  145 

7° 
•yft 

9.20  701 

0.79  299 

9-99  444 

51 

8 

10.7 

10.5 

10.4 

10.3 

10 

9.20  223 

7° 

9.20  782 

ol 

0.79  218 

9-99  442 

50 

9 

12.0 

11.8 

11.7 

n.6 

ii 

12 

9.20  302 
9.20  380 

79 

78 

9.20  862 
9.20  942 

oO 
80 

0.79  138 
0.79058 

9-99  440 
9-99  438 

49 

48 

10 

20 

13-3 
26.7 

13-2 
26.3 

13-0 
26.0 

12.8 

25-7 

13 

9.20458 

78 

9.21  O22 

80 

0.78978 

9-99  436 

47 

30 

40.0 

39-5 

39-o 

38.5 

14 
15 

9-20  535 
9.20  613 

77 
78 

9.21  IO2 
g.2I  l82 

80 
80 

0.78  898 
0.78  818 

9-99  434 
9-99  432 

46 
45 

40 
50 

53-3 
66.7 

52.7 
65.8 

52.0 
65.0 

5L3 
64.2 

16 

9.20  691 

78 

g.2I  26l 

79 

0.78  739 

9.99  429 

44 

17 

9.20  768 

77 

9.2I  341 

80 

0.78  659 

9.99427 

43 

fO 

(0 

74 

i& 

18 

9.20  845 

77 

9.21  420 

79 

0.78  580 

9-99  425 

42 

L3 

19 
20 

21 

9.20  922 

77 
77 
77 

9-21  499 

79 
79 
79 

0.78  501 

9-99  423 

40 

39 

3 
4 

5 

2.5 
3-8 
5.1 
6.3 

2.5 
3-8 
5-o 

6.2 

2-5 

3-7 
4-9 

6.2 

2.4 
3-6 

4-9 
6.1 

9.20  999 

9-21  578 

0.78  422 

9-99  42i 

9.21  076 

9-21  657 

0.78  343 

9.99419 

22 
23 

9-2i  153 
9.21  229 

77 
76 
77 

9-21  736 
9.21  814 

79 

78 
7Q 

0.78  264 
0.78  186 

9.99417 
9-994I5 

38 
37 

6 

7 

7.6 
8-9 

7-5 
8.8 

7-4 

8.6 

7-3 

8,5 

24 

9.21  306 

if* 

9-21  893 

t-Q 

0.78  107 

9.99413 

36 

8 

IO.I 

IO.O 

9-9 

9-7 

25 

9.21  382 

70 
76 

9.21971 

7° 
78 

0.78  029 

9.99411 

35 

9 

11.4 

II.  2 

ii.  i 

II.O 

26 

9.21  458 

/u 

76 

9.22  049 

IV 

78 

0.77951 

9.99  409 

34 

10 

12.7 

12.5 

12.3 

12.2 

27 
28 

9-21  534 
9.21  610 

/u 
76 

9.22  127 
9.22  205 

J° 

78 

_D 

o.77  873 
0-77  795 

9-99  407 
9-99  404 

33 

32 

20 
30 

25-3 
38.0 

25.0 
37-5 

24.7 
37-0 

24.3 
36.5 

29 

30 

9.21  685 

75 
76 

9.22  283 

7° 

78 

0.77717 

9-99  402 

30 

40 
50 

50.7 
63-3 

5O.O 
62.5 

49-3 
61.7 

48.7 
60.8 

9.21  701 

9.22  361 

0.77  639 

9.99  400 

31 

9.21  836 

/D 

9-22  438 

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0.77  562 

9-99  398 

29 

79 

71 

3 

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32 

9.21  912 

76 

9.22  516 

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77 

0.77  484 

9-99  396 

I 

1.2 

1.2 

o.o 

O.O 

33 

9.21  987 

75 

9-22  593 

77 

0.77407 

9-99  394 

27 

2 

2.4 

2-4 

O.I 

O.I 

34 

9.22  062 

9.22  670 

77 

0.77330 

9-99  392 

26 

3 

3.6 

3-6 

0.2 

O.I 

35 

9.22  137 

75 

9-22  747 

0.77253 

9-99  390 

25 

4 

4.8 

4-7 

0.2 

O.I 

36 

9.22  211 

74 

75 

9.22  824 

77 
77 

0.77  176 

9-99  388 

24 

5 

6.0 

5-9 

O.2 

0.2 

37 

9.22  286 

9.22  901 

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0.77  099 

9-99  385 

23 

6 

7-2 

7-1 

0.3 

0.2 

38 

9-22  361 

75 

9-22977 

76 

0.77023 

9-99  383 

22 

7 

8.4 

8-3 

0.4 

0.2 

39 

9-22435 

74 

9.23  054 

77 
76 

0.76  946 

9-99  38i 

21 

8 

9.6 

9-5 

0.4 

0-3 

40 

9-22  509 

7.1 

9-23  130 

76 

0.76  870 

9-99  379 

20 

9 

10.8 

10.6 

T  T  R 

0.4 

0.3 

42 
43 
44 
45 

9.22  583 
9-22  657 
9-22  731 
9.22  8o§ 

9.22878 

74 
74 
74 
73 
74 

9.23  206 
9.23  283 
9-23  359 

9-23  435 
9.23  510 

77 
76 
76 

75 
76 

0.76  794 
0.76717 
0.76  641 

0.76  565 
0.76  490 

9-99377 
9-99  375 
9-99  372 
9-99  370 
9-99  368 

19 

18 
17 
16 
15 

10 

20 
30 
40 
50 

24.0 
36.0 
48.0 
60.0 

23.7 

35-5 
47-3 
59.2 

0-5 
1.0 

1.5 
2.0 

2.5 

0-3 
0.7 
I.O 

i.7 

46 

9-22  952 

73 

9.23  586 

75 

0.76414 

9-99  366 

14 

47 

9.23025 

9.23  661 

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0.76  339 

9.99  364 

13 

333 

48 

9.23  098 

73 

9-23  737 

70 

0.76  263 

9-99  362 

12 

49 

9.23  171 

73 

9.23  812 

75 

0.76  188 

9-99  359 

II 

79   78  77 

50 

9.23  244 

/  j 

9-23  887 

7 

0.76  113 

9-99357 

10 

°  13.2  13.0  12.8 

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9.233I7 

73 

9.23  962 

7~ 

0.76038 

9.99355 

9 

39.5  39.0  38.5 

52 

9.23  390 

73 

9.24037 

75 

0-75  963 

9-99353 

8 

65.8  65.0  64.2 

53 

9.23  462 

73 

9.24  112 

75 
74 

0.75  888 

9.99351 

7 

54 

9-23  535 

9.24  186 

0.75  814 

9.99  348 

6 

30 

55 

9.23  607 

7* 

9.24  261 

75 

o.75  739 

9-99  346 

5 

3 

56 

9.23  679 

72 

73 

9-24335 

74 

75 

0.75  665 

9-99  344 

4 

76  75 

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57 
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9-23  752 
9.23  823 
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72 

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9-24  4$p 
9.24  4«4 
9.24  558 

74 
74 

0.75  590 
0.75  516 
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9.99  342 
9.99  340 
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9.23  967 

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0.75  368 

9-99  335 

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*170°  260°  *350C 


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L  Tan 

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9.23  967 

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0.75  368 

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9.24  039 
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9.24  181 

9-24  253 
9-24  324 
9-24  395 
9.24  466 

9-24  536 
9.24  607 

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71 
72 

7i 
71 
71 
70 

71 

70 

9.24  706 
9-24  779 
9-24  853 
9.24  926 
9.25  ooo 
9-25  073 
9-25  146 
9.25  219 
9-25  292 

74 

73 
74 
73 
74 
73 
73 
73 
73 

T\ 

0.75  294 

0.75  221 
0-75  147 

o-75  074 
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0.74927 
0.74  854 
0.74  781 
0.74  708 

9-99  333 
9-99331 
9-99  328 
9-99  326 
9-99  324 
9-99  322 

9-99  319 
9-99317 
9-993I5 

2 

3 

2 

2 

2 

3 

2 

2 
2 

59 

58 
57 
56 
55 
54 

53 
52 
5i 

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2 

3 
4 

5 
6 

7 
8 

74 

1.2 
2-5 

3-7 
4-9 

6.2 

7-4 
8.6 

9-9 

73 

1.2 
2.4 

3-6 
4-9 
6.1 
7-3 
8-5 
9-7 

72 

1.2 
2.4 

3-6 

4-8 
6.0 

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8.4 
9.6 

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10 

9.24677 

71 

9-25  365 

0.74635 

9.99313 

50 

9 

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12 

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15 
16 

17 

18 

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9.24  748 
9.24818 
9.24  888 
9.24958 
9.25  028 
9-25  098 
9.25  168 
9-25  237 
9-25  307 

70 
70 
70 

70 
70 
70 
69 
70 
60 

9-25  437 
9-25  5io 
9-25  582 

9-25  655 
9.25  727 

9-25  799 
9-25  871 
9-25  943 
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73 
72 
73 
72 
72 
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0.74  490 
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0.74273 
0.74  201 
0.74  129 
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9-99  3io 
9.99  308 
9-99  306 
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9-99  3oi 
9-99  299 
9-99  297 
9-99  294 
9.99  292 

2 
2 
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3 

2 
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49 
48 
47 
46 
45 
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43 
42 

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10 

20 

30 
40 
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70 

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1.2 

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20 

9-25  376 

60 

9.26086 

71 

72 

0.73  914 

9.99  290 

40 

4 

4-7 

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21 
22 
23 
24 
25 
26 

27 
28 
29 

9-25  445 
9-25  5U 
9-25  583 
9-25  652 
9-25  72i 
9.25  790 

9-25  858 
9.25  927 
9-25  995 

69 
69 
69 
69 
69 
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69 
68 
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9.26158 
9.26  229 
9.26  301 

9-26372 
9.26  443 
9.26  514 
9-26585 
9.26655 
9.26  726 

71 
72 
71 
71 
71 
71 
70 
71 

0.73  842 
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0-73  699 
9.73  628 

0-73  557 
0.73  486 

0.73415 
0-73  345 
0.73  274 

9.99  288 
9-99  285 
9-99  283 
9.99  281 
9-99  278 
9-99  276 

9-99  274 
9.99271 
9.99  269 

3 

2 
2 

3 

2 
2 

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39 
38 
37 
36 
35 
34 

33 
32 
31 

6 

7 
8 

9 

10 
20 

30 
40 

5-9 
7-i 
8.3 
9-5 
10.6 

n.8 
23.7 
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47-3 

5-8 
7-o 

8.2 

9-3 
10.5 

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9.92  227 
9-92  253 
9-92  279 
9.92  304 
9.92  330 
9-92  356 

26 
26 
25 
26 
26 
25 
26 
26 
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0.07  850 
0.07  824 
0.07  798 
0.07  773 
0.07  747 
0.07  721 
0.07  696 
0.07  670 
0.07  644 

9.88521 
9.88  510 
9.88  499 
9.88489 
9.88478 
9.88  468 
9.88457 
9-88  447 
9.88436 

II 
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IO 

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9 

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7 
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3 
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5 
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8 
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22.5 
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12.5 
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19-3 
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9.80  807 

9.92  381 

0.07  619 

9-88425 

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9.80  822 
9.80837 
9.80  852 
9.80  867 
9.80  882 
9.80  897 
9.80912 
9.80927 
9.80  942 

15 
15 
15 
15 
15 
15 
15 
15 

9-92  407 
9-92  433 
9.92  458 

9.92  484 
9-92  5io 
9-92  535 
9.92  561 
9.92  587 
9.92  612 

26 

25 
26 

26 

25 
26 

26 

25 

0.07  593 
0.07  567 
0.07  542 
0.07  516 
0.07  490 
0.07  465 
0.07  439 
0.07413 
0.07  388 

9.88415 
9.88  404 
9.88  394 

9-88383 
9-88372 
9.88  362 

9-88351 
9.88  340 
9.88  330 

II 

10 

II 
II 

10 

II 
II 

IO 
J  J 

59 

58 
57 
56 
55 
54 

53 
52 
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3 
4 

5 
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7 
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2.2 
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3-5 
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2-5 

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9.80957 

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9.92  638 

20 

0.07  362 

9.88319 

1  J 

50 

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12 
13 
14 

15 

16 

17 
18 

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9.80  972 
9.80987 

9.81  002 
9.81  017 
9.81  032 
9.81  047 

9.81  061 
9.81  076 
9.81  091 

15 
15 
15 
15 
15 
15 
14 
15 
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9.92  663 
9.92  689 
9.92  715 
9.92  740 
9.92  766 
9.92  792 
9.92817 
9.92  843 
9.92  868 

25 
26 
26 
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26 
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25 
26 
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0.07  337 
0.07  311 
0.07  285 
0.07  260 
0.07  234 
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0.07  183 
0.07  157 
0.07  132 

9.88  308 
9.88  298 
9.88287 
9.88276 
9.88  266 
9.88255 
9.88  244 
9.88  234 
9.88  223 

10 

II 
II 

IO 

II 
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49 
48 
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46 
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43 
42 
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20 
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40 
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3 
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9.81  106 

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9.92  894 

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0.07  1  06 

9.88  212 

40 

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21 
22 
23 
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25 
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28 

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9.81  121 

9.81  136 
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9.81  180 
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9-93  099 
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9.88  201 
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9.88  180 

9.88  169 
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9-88  137 
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9.88  105 

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9.81  269 
9.81  284 

9.81  299 

9.81  314 
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9-  8  1  343 
9-81  358 
9.81  372 
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15 
15 
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9-93  201 
9.93  227 
9.93  252 
9.93  278 
9-93  303 
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9.88  094 
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9-81  563 
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14 
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9-93  687 
9.93  712 
9-93  738 
9-93  763 
9-93  789 
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9-93  840 
9.93  865 
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25 
26 

25 
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25 
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25 
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0.06313 
0.06  288 
0.06  262 
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0.06  135 
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9.87  877 
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9-87855 
9.87  844 

9-87  833 
9.87  822 

9.87  811 
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0.06  084 

9.87  778 

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12 
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12 
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II 
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12 
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12 
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12 
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12 
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60 

59 

58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 

48 

47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
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33 
32 
31 
30 
29 
28 
27 
26 
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24 
23 

22 
21 
20 
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18 
17 
16 
15 
14 
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10 

9 

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7 
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10 
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26 
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9.81  709 
9.81  723 
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9.81  752 
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9.81  781 
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14 
15 
14 
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9.93  942 
9.93  967 
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9-94  044 
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9.94  1  20 
9-94  146 

0.06  058 
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0.05  982 
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9.87  767 
9.87  756 

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9.87  734 
9.87  723 
9.87712 
9.87  701 
9.87  690 
9.87  679 

9-81  839 

14 
15 
14 
14 
15 
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9.94171 

0.05  829 

9.87,668 

9.81  854 
9.81  868 
9.81  882 
9.81  897 
9.81  911 
9.81  926 
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9-8i  955 
9.81  969 

9.94  197 

9.94  222 
9-94  248 

9-94  273 
9-94  299 
9«94  324 
9-94  350 
9-94375 
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0.05  803 
0.05  778 
0.05  752 
0.05  727 
0.05  701 
0.05  676 
0.05  650 
0.05  625 
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9.87657 
9.87  646 
9-87  635 
9.87  624 
9-87613 
9.87  601 

9.87  590 
9.87  579 
9.87  568 

9.81  983 

14 

9-94  426 

0-05  574 

9.87557 

9.81  998 
9.82012 
9.82  026 
9.82041 
9-82055 
9.82  069 

9.82  084 
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9.82  112 

15 
14 
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9-94452 
9-94477 
9-94  503 
9-94  528 
9-94  554 
9-94  579 
9.94  604 
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0.05  548 
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0.05  396 
0.05  370 
0.05  345 

9.87  546 

9-87  535 
9.87  524 

9.87513 
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9.87  490 

9.87  479 
9.87468 

9.87457 

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14 

9.94681 

0.05  319 

9.87446 

31 
32 

33 
34 
35 
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37 
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9.82  141 

9.82155 
9.82  169 

9.82  184 
9.82  198 
9.82  212 
9.82  226 
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15 
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9-94  706 
9-94  732 
9-94  757 

9-94  783 
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9-94  834 

9-94  859 
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0.05  294 
0.05  268 
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9.87434 
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9.87412 

9.87401 
9.87  390 
9.87  378 
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9.87  345 

40 

9.82  269 

14 
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9.87  334 

41 
42 
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9.82  283 
9.82  297 
9.82311 
9.82  326 
9.82  340 

9.82  354 
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9.82  382 
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9-94  961 
9.94  986 
9-95  012 
9-95  037 
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9-95  139 
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9.87  322 
9.87311 
9-87  300 
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9-^7  277 
9.87  266 

9.87255 
9.87  243 
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9.82  410 

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53 
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9.82  424 
9.82  439 
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9-82  509 
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9-95  215 
9-95  240 
9-95  266 
9.95  291 
9.95317 
9-95  342 
9-95  368 
9-95  393 
9.95  418 

25 

25 
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0.04  760 
0.04  734 
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0.04  632 
0.04  607 
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9.87  209 
9.87  198 
9.87  187 

9.87i75 
9.87  164 

9.87  153 
9.87  141 
9.87  130 
9.87  119 

60 

9.82  551 

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9.95  444 

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9.87  107 

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9.82  579 
9-82  593 
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14 
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9-95  469 
9-95  495 
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9-95  545 
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9-95  596 
9.95  622 
9-95  647 
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0.04  353 
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9.87  096 
9.87085 
9.87073 
9.87  062 
9.87050 
9.87  039 
9.87028 
9.87  016 
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12 
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12 
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57 
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9.82  691 

9-95  698 

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0.04  302 

9.86  993 

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13 
14 
15 
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9.82  705 
9.82  719 
9-82  733 
9.82  747 
9.82  761 
9.82  775 
9.82  788 
9.82  802 
9.82816 

14 
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9-95  723 
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9.84  656 

13 

9.99419 

o.oo  581 

9-85  237 

I3 

23 

4 

9.0 

8 

•7 

38 

9.84  669 

*3 

9-99  444 

25 

o.oo  556 

9-85  225 

12 

22 

5 

II.O 

10.6 

39 

9.84  682 

9.99  469 

25 
26 

o.oo  531 

9.85  212 

L^> 

21 

7 

13-0 

12.5 

40 

9.84  694 

9-99  495 

o.oo  505 

9.85  200 

20 

8 

15.0 

14.4 

42 
43 

9.84  707 
9.84  720 
9.84  733 

1  j 
13 
13 

12 

9.99  520 
9-99  545 
9«99  570 

25 
25 
26 

0.00480 

0.00455 

o.oo  430 

9.85  187 
9.85  175 
9.85  162 

13 

12 

13 
12 

19 
18 
i? 

9 

10 

ii 

17.0 
ig.O 
21.0 

16.3 
18.3 

2O.2 
22  I 

44 

45 

9-84  745 
9.84  758 

13 

9-99  596 
9.99  621 

25 

o.oo  404 
o.oo  379 

9.85  150 
9.85  137 

13 

16 
15 

12 
13 

25.0 

24.1 

46 

9.84  771 

13 

9.99  646 

25 
26 

o.oo  354 

9.85  I2§ 

12 

TO 

14 

12 

12 

47 

9.84  784 

9.99  672 

o.oo  328 

9.85  112 

13 

9A 

\ 

J5 

48 

9.84  796 

9.99  697 

25 

o.oo  303 

9.85  100 

12 

12 

49 

9.84  809 

I3 

9.99  722 

25 

o.oo  278 

9.85087 

1J 

II 

I. 

I 

i 

.1 

50 

9.84  822 

9-99  747 

26 

o.oo  253 

9.85  074 

13 

10 

2 

3-2 

3-1 

52 
53 

9-84  835 
9.84  847 
9.84  860 

1  j 

12 

13 

9-99  773 
9.99  798 
9.99  823 

25 
25 
25 

o.oo  227 

O.OO  2O2 

o.oo  177 

9.85  062 
9.85  049 
9.85037 

13 

12 

13 

9 

8 

7 

3 
4 
5 

5.4 

7-6 
9.8 

5-2 

7.3 
9.4 

1  1  2 

54 

55 
56 

9.84  873 
9.84885 
9.84  898 

12 

13 
13 

9.99  848 
9.99874 
9.99  899 

26 

25 
25 

o.oo  152 
o.oo  126 

0.00  101 

9.85  024 
9.85012 

9.84  999 

12 
13 

6 

5 
4 

6 

7 
8 

14.1 

1  6.2 

18.4 

15^6 
17-7 

57 

9.84911 

9-99  924 

o.oo  076 

9.84  986 

12 

3 

9 

20.6 

19.8 

58 

9.84  923 

9.99  949 

25 

•7  ft 

0.00051 

• 

T  <i 

2 

10 

22.8 

21 

•9 

59 

9.84  936 

13 

9-99  975 

0.00025 

9.84  961 

12 

I 

1  1 

24.9 

23.9 

60 

M-<M  <)4<> 

O.I  X  XXX) 

0.00000 

.,.84  ,,49 

0 

LCos 

d 

LCot 

cd 

L  Tan 

LSin 

d 

' 

P  P 

*135°  225°  *315° 


TABLE  OF  THE  NATURAL 

TRIGONOMETRIC  FUNCTIONS 


FROM  MINUTE  TO  MINUTE. 


108 

*90°     180° 


*270e 


NATURAL 


*91°    181°     *27r 


' 

Sin 

Tan 

Cot 

Cos 

0 

o.oooo 

0.0000 

00 

.0000 

60 

I 

2 

3 
4 

6 

7 
8 

9 

0.0003 
0.0006 
0.0009 

O.OOI2 
0.0015 
0.0017 
0.0020 
0.0023 
O.OO26 

0.0003 
0.0006 
0.0009 

O.OOI2 

0.0015 
0.0017 

0.0020 
0.0023 
O.OO26 

3437.75 
1718.87 

1145.92 

859-436 
687.549 
572.957 

491.106 

429.718 
381.971 

.0000 

.0000 

.0000 
.0000 
.0000 

.0000 
.0000 

.0000 
.0000 

59 

58 

57 
56 
55 
54 

53 
52 
5i 

10 

O.OO29 

O.OO29 

343-774 

.0000 

50 

ii 

12 
13 
14 

ID 
16 

17 

18 

19 

0.0032 
0.0035 
O.OO38 
O.OO4I 
0.0044 
0.0047 
O.OO49 
0.0052 
0.0055 

0.0032 
0.0035 
0.0038 
O.OO4I 
0.0044 
0.0047 
0.0049 
O.OO52 
0.0055 

312.521 
286.478 
264.441 

245.552 
229.182 
214.858 
202.219 
190.984 
180.932 

.0000 
.0000 
I.OOOO 
I.OOOO 
I.OOOO 
I.OOOO 
I.OOOO 
.0000 

.0000 

49 

48 

47 
46 
45 
44 

43 
42 
4i 

20 

0.005  8  x 

0.0058 

171.885 

.0000 

40 

21 

22 
23 
24 
25 
26 

27 
28 

29 

0.0061 
0.0064 
0.0067 
0.0070 
0.0073 
0.0076 
0.0079 
0.0081 
0.0084 

O.OOGl 
0.0064 
O.OO67 
O.OO7O 

0.0073 
0.0076 
0.0079 
0.0081 
0.0084 

163.700 
156.259 
149.465 

143.237 
137.507 
132.219 

127.321 
122.774 
118.540 

.0000 
.0000 

.0000 
.0000 
.0000 

.0000 
.0000 
.0000 
.0000 

39 
38 
37 
36 
35 
34 
33 
32 
3i 

30 

0.0087 

0.0007 

114.589 

.0000 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 

0.0090 
0.0093 
0.0096 
0.0099 

O.OIO2 
O.OIO§ 
0.0108 
O.OIII 
O.OII3 

0.0090 
0.0093 
0.0096 
0.0099 

0.0102 
0.0105 
0.0  1  08 
O.OIII 
O.OII3 

110.892 
107.426 
104.171 
101.107 
98.2179 
95-489! 
92.9085 
90.4633 
88.1436 

.0000 

.0000 

I.OOOO 
I.OOOO 

0.9999 
0.9999 
0.9999 
0.9999 
0.9999 

29 

28 

27 
26 
25 
24 
23 

22 
21 

40 

O.OII6 

O.OII6 

85.9398 

0.9999 

20 

41 
42 

43 

44 
45 
46 

47 
48 
49 

O.OIig 
0.0122 
0.0125 
0.0128 
O.OI3I 
0.0134 
0.0137 
O.OI4O 
0.0143 

O.OII9 
0.0122 
0.0125 
0.0128 
O.OI3I 
0.0134 
0.0137 
O.OI4O 
0.0143 

83-S435 
81.8470 

79-9434 
78.1263 
76.3900 
74.7292 

73.1390 
71.6151 
70.1533 

0.9999 
0.9999 
0.9999 
0.9999 
0.9999 
0.9999 
0.9999 
0.9999 

0.9999 

19 
17 

16 

15 
14 
13 

12 
II 

oO 

0.0145 

0.0145 

68.7501 

0.9999 

10 

5i 
52 
53 
54 
55 
56 

57 
58 
59 

0.0148 
O.OI5I 
0.0154 
0.0157 

0.0160 
0.0163 
0.0166 
0.0169 

0.0172 

0.0148 
O.OI5I 
0.0154 
0.0157 

0.0160 
0.0163 
0.0166 
0.0169 
o.oi  72 

67.4019 
66.1055 
64.8580 
63.6567 
62.4992 
61.3829 
60.3058 

59-2659 
58.2612 

0.9999 
0.9999 
0.9999 
0.9999 
0.9999 
0.9999 
0.9999 
0.9999 

0.9999 

9 

8 

7 
6 
5 
4 
3 

2 

00 

ooi_75 

0.017^ 

"•'-'00' 

<  >.')'}<>*> 

0 

Cos 

Cot 

Tan 

Sin 

' 

>179e     260°     * 


NATURAL 


' 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 
4 
5 
6 

8 
9 

10 
ii 

12 
13 
14 

15 

1  6 

17 
18 

19 
20 

21 
22 
23 
24 

25 
26 

27 

28 

29 

30 

3i 
32 
33 
34 
35 
36 

37 
38 

40 

4i 

42 
43 
44 
45 
46 

47 
48 

49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 
00 

0.0175 

0.0175 

57.2900 

0.9998 

60 

59 
58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 

47 
46 
45 
44 

43 
42 
4i 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 

28 

27 
26 
25 
24 
23 

21 

20 

19 

18 

17 
16 

15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 
I 

0 

0.0177 
o.oi  So 
0.0183 
o.oi  86 
0.0189 
0.0192 
0.0195 
0.0198 

O.O2OI 

0.0177 
o.oi  So 
0.0183 
o.oi  86 
0.0189 
0.0192 
0.0195 
0.0198 

O.02OI 

56.3506 
55.4415 
54-5613 
53-7086 
52.8821 
52.0807 

51-3032 
50.5485 
49-8i57 

0.9998 
0.9998 
0.9998 
0.9998 
0.9998 
0.9998 
0.9998 
0.9998 
0.9998 

O.O2O4 

0.0204 

49.1039 

0.9998 

O.O2O7 
O.O2O9 
0.0212 
0.0215 

0.0218 

O.O22I 
O.O224 
O.O227 
O.O23O 

0.0207 
0.0209 

O.O212 
O.O2I5 
0.0218 
O.O22I 
0.0224 
O.O227 
O.O23O 

48.4121 

47-7395 
47-0853 

46.4489 
45-8294 
45.2261 

44.6386 
44.0661 
43.5081 

0.9998 
0.9998 
0.9998 
0.9998 
0.9998 
0.9998 
0.9997 
0.9997 
0.9997 

0.0233 

0.0233 

42.9641 

0.9997 

0.0236 
0.0239 
0.0241 
0.0244 
0.0247 
O.O25O 
0.0253 
0.0256 
O.O259 

0.0236 
0.0239 
O.O24I 
O.O244 
0.0247 
O.O25O 
0.0253 
0.0256 
O.O259 

42-4335 
41-9158 
41.4106 

40.9174 

40.4358 
39-9655 

39.5059 
39-0568 
38.6177 

0.9997 
0.9997 
0.9997 
0.9997 
0.9997 
0.9997 
0.9997 
0.9997 
0.9997 

O.O262 

0.0262 

38.1885 

0-9997 

O.O265 
0.0268 
0.0270 
0.0273 
O.O276 
O.O279 
O.O282 
0.0285 
0.0288 

0.026§ 
O.O268 
O.O27I 
O.O274 
0.0276 
0.0279 
0.0282 
0.0285 
0.02SS 

37.7686 

37-3579 
36.9560 

36.5627 
36.1776 
35-Soo6 

354313 
35-0695 
34.7I5I 

0.9996 
0.9996 
0.9996 
0.9996 
0.9996 
0.9996 
0.9996 
0.9996 
0.9996 

0.0291 

O.O29I 

34o678 

0.9996 

0.0294 
O.O297 
O.O3OO 
O.O3O2 
0.0305 
O.O3O8 
0.031  1 
0.0314 
0.0317 

0.0294 
0.0297 
0.0300 
0.0303 
0.0306 
0.0308 
0.03II 
0.0314 
0.0317 

34-0273 

33.6935 
33.3662 

33.0452 
32.7303 
32.4213 
32.1181 
31.8205 
31.5284 

0.9996 
0.9996 
0.9996 
0.9995 
0-9995 
0-9995 

0-9995 
0.9995 

o.<)o<)5 

O.O32O 

0.032O 

31.2416 

0.9995 

0.0323 
0.0326 
0.0329 
0.0332 
0.0334 
0.0337 

0.0340 
0.0343 
CXO346 

0.0323 
0.0326 
0.0329 
0.0332 
0.0335 
0.0338 
0.0340 

0.0343 
0.0346 

30.9599 
30.6833 
30.4116 
30.1446 
29.8823 
29.6245 
29.3711 
29.1220 
28.8771 

0-9995 
0.9995 
0.9995 
0.9995 
0.9994 
0.9994 

0.9994 
0.9994 
0.9994 

O.O349 

o.o;  )-) 

•_"v  (.-,<)•; 

•'.•IT,  I 

Cos 

Cot 

Tan 

Sin 

*178°    268°    *358« 


*92°  182°  *272C 


NATURAL 


*93C 


109 
183°  *273° 


' 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 
4 
5 
6 

7 
8 

9 
10 
ii 

12 
13 
14 
15 

16 

17 

18 

19 
20 

21 
22 
23 

24 
25 
26 

27 
28 
29 

30 

31 
32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 
42 
43 
44 
45 
46 

47 
48 
49 
50 

5i 

52  . 
53 
54 
55 
56 

57 
58 
59 
60 

0.0349 

0.0349 

28.6363 

0.9994 

60 

59 

58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 
47 
46 
45 
44 

43 
42 
4i 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 
29 
28 
27 
26 

25 
24 

23 

22 
21 
20 
19 

18 
17 
16 
15 
14 
13 

12 

II 
10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.0352 
0.0355 
0.0358 
0.0361 
0.0364 
0.0366 
0.0369 
0.0372 
0.0375 

0.0352 
0.0355 
0.0358 
0.0361 
0.0364 
0.0367 
0.0370 
0.0373 
0.0375 

28.3994 
28.1664 
27.9372 
27.7117 
27-4899 
27-2715 
27.0566 
26.8450 
26.6367 

0.9994 
0.9994 
0.9994 

0.9993 
0.9993 
0-9993 
0.9993 
0.9993 
0.9993 

0.0378 

0.0378 

26.4316 

0.9993 

0-9993 
0.9993 
0.9993 
0.9992 
0.9992 
0.9992 
0.9992 
0.9992 
0.9992 

0.0381 
0.0384 
0.0387 
0.0390 
0.0393 
0.0396 
0.0398 
0.0401 
0.0404 

0.0381 
0.0384 
0.0387 
0.0390 
0.0393 
0.0396 
0.0399, 
0.0402 
0.0405 

26.2296 
26.0307 
25.8348 
25.6418 

25-4517 
25.2644 

25.0798 
24.8978 
24.7185 

0.0407 

0.0407 

24.5418 

0.9992 

0.0410 
0.0413 
0.0416 
0.0419 
0.0422 
0.0425 
0.0427 
0.0430 
0.0433 

0.0410 
0.0413 
0.0416 
0.0419 
0.0422 
0.0425 
0.0428 
0.0431 
0.0434 

24-3675 
24.1957 
24.0263 

23.8593 
23-6945 
23-5321 
23.3718 
23.2137 
23-0577 

0.9992 
0.9991 
0.9991 
0.9991 
0.9991 
0.9991 
0.9991 
0.9991 
0.9991 

0.0436 

0.0437 

22.9038 

0.9990 

0.0439 
0.0442 
0.0445 
0.0448 
0.0451 
0.0454 
0.0457 
0.0459 
0.0462 

0.0440 
0.0442 
0.0445 
0.0448 
0.0451 
0.0454 
0.0457 
0.0460 
0.0463 

22.7519 
22.6020 
22.4541 
22.3081 
22.1640 
22.0217 

21.8813 
21.7426 
21.6056 

0.9990 
0.9990 
0.9990 
0.9990 
0.9990 
0.9990 
0.9990 
0.9989 
0.9989 

0.0465 

0.0466 

21.4704 

0.9989 

0.0468 
0.0471 
0.0474 
0.0477 
0.0480 
0.0483 
0.0486 
0.0488 
0.0491 

0.0469 
0.0472 
0.0475 
0.0477 
0.0480 
0.0483 
0.0486 
0.0489 
0.0492 

21.3369 
21.2049 
21.0747 
20.9460 
20.8188 
20.6932 
20.5691 
20.4465 
20.3253 

0.9989 
0.9989 
0.9989 
0.9989 
0.9988 
0.9988 
0.9988 
0.9988 
0.9988 

0.0494 

0.0495 

20.2056 

0.9988 

0.0497 
0.0500 
0.0503 
0.0506 
0.0509 
0.0512 
0.0515 
0.0518 
0.0520 

0.0498 
0.0501 
0.0504 
0.0507 
0.0509 
0.0512 
0.0515 
0.0518 
0.0521 

20.0872 
19.9702 
19-8546 
19.7403 
19.6273 
19-5156 
19.4051 
19.2959 
19.1879 

0.9988 
0.9987 
0.9987 
0.9987 
0.9987 
0.9987 
0.9987 
0.9987 
0.9986 

0.0523 

0.0524 

19.0811 

0.9986 

Cos  1  Cot 

Tan 

Sin 

' 

267°  *357°       87' 


' 

Sin 

Tan 

Cot 

Cos 

0 

0.0523 

0.0524 

19.0811 

0.9986 

60 

I 

2 

3 

4 

6 

8 
9 

0.0526 
0.0529 
0.0532 
0.0535 
0.0538 
0.0541 
0.0544 
0.0547 
0.0550 

0.0527 
0.0530 
0.0533 
0.0536 
0.0539 
0.0542 
0.0544 
0.0547 
0.0550 

18.9755 
18.8711 
18.7678 
18.6656 
18.5645 
18.4645 
18.3655 
18.2677 
18.1708 

0.9986 
0.9986 
0.9986 
0.9986 
0.9986 
0.9985 
0.9985 
0.9985 
0.9985 

59 

58 
57 
56 
55 
54 

53 
52 
5i 

10 

0.0552 

0.0553 

18.0750 

0.9985 

50 

ii 

12 
13 
14 
15 

16 

17 

18 

19 

0.0555 
0.0558 
0.0561 
0.0564 
0.0567 
0.0570 
0.0573' 
0.0576 
0.0579 

0.0556 
0.0559 
0.0562 
0.0565 
0.0568 
0.0571 
0.0574 
0.0577 
0.0580 

17.9802 
17.8863 
17-7934 
17.7015 
17.6106 
17.5205 

174314 
17-3432 

17.2558 

0.9985 
0.9984 
0.9984 
0.9984 
0.9984 
0,3984 
0.9984 
0.9983 
0.9983 

49 

48 
47 
46 
45 
44 

43 
42 
41 

20 

0.0581 

0.0582 

17.1693 

0.9983 

40 

21 

22 
23 
24 
25 
26 

27 

28 
29 

0.0584 
0.0587 
0.0590 
0.0593 
0.0596 
0.0599 
0.0602 
0.0605 
0.0608 

0.0585 
0.0588 
0.0591 
0.0594 
0.0597 
0.0600 
0.0603 
0.0606 
0.0609" 

17.0837 
16.9990 
16.9150 
16.8319 
16.7496 
1  6.668  1 
16.5874 
16.5075 
16.4283 

0.9983 
0.9983 
0.9983 
0.9982 
0.9982 
0.9982 
0.9982 
0.9982 
0.9982 

39 
38 
37 
36 
35 
34 

33 
32 
31 

30 

0.0610 

0.0612 

16.3499 

0.9981 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 

0.0613 
0.0616 
0.0619 
0.0622 
0.0625 
0.0628 
0.0631 
0.0634 
0.0637 

0.0615 
0.0617 
0.0620 
0.0623 
0.0626 
0.0629 
0.0632 
0.0635 
0.0638 

16.2722 
16.1952 
16.1190 
16.0435 
15.9687 
15.8945 
15.8211 

15.7483 
15.6762 

0.9981 
0.9981 
0.9981 
0.9981 
0.9980 
0.9980 
0.9980 
0.9980 
0.9980 

29 
28 

27 
26 
25 
24 
23 

22 
21 

40 

0.0640 

0.0641 

15.6048 

0.9980 

20 

4i 
42 

43 
44 
45 
46 

47 

48 

49 

0.0642 
0.0645 
0.0648 
0.0651 
0.0654 
0.0657 
0.0660 
0.0663 
0.0666 

0.0044 
0.0647 
0.0650 
0.0653 
0.0655 
0.0658 
0.0661 
0.0664 
0.0667 

15.5340 
15.4638 
15.3943 
15.3254 
15.2571 
15.1893 

15.1222 

15.0557 
14.9898 

0.9979 
0.9979 
0.9979 

0.9979 
0.9979 
0.9978 
0.9978 
0.9978 
0.9978 

18 
17 
16 
15 
14 
13 

12 
II 

50 

0.0669 

0.0670 

14.9244 

0.9978 

10 

5i 
52 
53 
54 
55 
56 

57 
58 
59 

0.0671 
0.0674 
0.0677 
0.0680 
0.0683 
0.0686 
0.0689 
0.0692 
0.0695 

0.0673 
0.0676 
0:0679 
0.0682 
0.0685 
0.0688 
0.0690 
0.0693 
0.0696 

14.8596 
14.7954 
I4.73I7 
14.6685 
14.6059 
I4.5438 
14.4823 
14.4212 
14.3607 

0-9977 
0.9977 
0.9977 
0.9977 
0.9977 
0.9976 
0.9976 
0.9976 
0.9976 

9 

8 

7 
6 
5 
4 
3 

2 

I 

60 

0.0698 

0.0699 

14.3007 

0.9976 

0 

Cos 

Cot 

Tan 

Sin 

1 

NATURAL 


86< 


*176°  266°  *356° 


110 
*94°     184'    *274°  4° 


NATURAL 


*95°     185°     *2T5 


' 

Sin 

Tan 

Cot 

Cos 

0 

0.0698 

0.0699 

14.3007 

0.9976 

GO 

59 
58 
57 
56 
55 
54 
53 
52 
5i 
50 

49 

48 

47 
46 
45 
44 

43 
42 
4i 
40 

39 
3S 
37 
36 
35 
34 

33 
32 
3i 
30 

I 

2 

3 
4 

6 

8 
9 
10 
ii 

12 
13 
14 
15 

16 

17 
18 

19 
20 

21 
22 
23 

24 
25 
26 

27 
28 

29 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 

42 
43 
44 
45 
46 

47 
48 
49 
50 

5i 

52 
53 
54 
55 
56 

57 
58 
59 
60 

0.0700 
0.0703 
0.0706 
0.0709 
0.0712 
0.0715 
0.0718 
0.0721 
0.0724 

0.0702 
0.0705 
0.0708 
0.0711 
0.0714 
0.0717 
0.0720 
0.0723 
0.0726 

14.2411 
14.1821 

14-1235 
14.0655 
14.0079 
13-9507 
13.8940 
13.8378 

13-7821 

0.9975 
0.99/5 
0.9975 
0.9975 
0.9975 
0.9974 

0.99/4 
0.9974 
0.9974 

0.0727 

0.0729 

13.7267 

0.9974 

0.0729 
0.0732 
0.0735 
0.0738 
0.0741 
0.0744 
0.0747 
0.0750 
0.0753 

0.0731 
0.0734 
0.0737 
0.0740 
0.0743 
0.0746 
0.0749 
0.0752 
0.0755 

13.6719 
13.6174 
13.5634 
13-5098 
13.4566 

13-4039 
13.3515 
13.2996 
13.2480 

0.99/3 
0-9973 
0-9973 
0-9973 
0-9973 
0.9972 

0.9972 
0.9972 
0.9972 

0.0756 

0.0758 

13.7969 

0.9971 

0.0758 
0.0761 
0.0764 
0.0767 
0.0770 
0.0773 
0.0776 
0.0779 
0.0782 

0.0761 
0.0764 
0.0767 
0.0769 
0.0772 
0.0775 
0.0778 
0.0781 
0.0784 

13.1461 
13.0958 
13.0458 

12.9962 

12.9469 

12.8981 
12.8496 
12.8014 

*i2.753f 

0.9971 
0.9971 
0.9971 
0.9971 
0.9970 
0.9970 
0.9970 
0.9970 
0.9969 

0.0785 

0.07157 

12.7062 

0.9969 

0.0787 
0.0790 
0.0793 
0.0796 
0.0799 
0.0802 
0.0805 
0.0808 
0.0811 

0.0790 
0.0793 
0.0796 
0.0799 
0.0802 
0.0805 
0.0808 
0.0810 
0.0813 

12.6591 

1  2.  6  1  2.] 

12.5660 
12.5199 
12.4742 

12.4288 

12.3838 
12.3390 

12.2946 

0.9969 
0.9969 
0.9968 
0.9968 
0.9968 
0.9968 
0.9968 
0.9967 
0.9967 

29 
28 

27 
26 

25 
24 

23 

22 
21 

20 

19 

18 
17 
16 
15 
M 
13 

12 
I  I 

0.0814 

0.08  1  6 

12.2505 

0.9967 

0.0816 
0.0819 
0.0822 
0.0825 
0.0828 
0.0831 
0.0834 
0.0837 
0.0840 

0.0819 
0.0822 
0.0825 
0.0828 
0.0831 
0.0834 
0.0837 
0.0840 
0.0843 

12.2067 
12.1632 

12.1201 
12.0772 

I2.034( 

11.9923 

11.9504 
11.9087 
11.8673 

0.9967 
0.9966 
0.9966 
0.9966 
0.9966 
0.9965 
0.9965 
0.9965 
0.9965 

0.0843 

0.0846 

11.8262 

0.9964 

10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.0845 
0.0848 
0.0851 
0.0854 
0.0857 
0.0860 
0.0863 
0.0866 
0.0869 

0.0849 
0.0851 
0.0854 
0.0857 
0.0860 
0.0863 
0.0866 
0.0869 
0.0872 

11.7853 
11.7448 
11.7045 
11.6645 

11.624$ 

11.5853 
11.5461 
11.5072 
11.4685 

0.9964 
0.9964 
0.9964 
0.9963 
0.9963 
0.9963 

0.9963 
0.9962 
03363 

0.0872 

o.o~7; 

1  !.|  -.01 

C..0.,'.2 

Cos 

Cot 

Tan 

Sin 

' 

*175°     265°     *355° 


NATURAL 


' 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 
4 

6 

7 

8 

9 

0.0872 

0.0875 

11.4301 

0.9962 

60 

59 
5S 
57 
56 
55 
54 

53 
52 
51 
50 

49 

48 
47 
46 
45 
44 

43 
42 
4i 
40 

39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 
25 
24 
23 

22 
21 

0.0874 
0.0877 
0.0880 
0.0883 
0.0886 
0.0889 
0.0892 
0.0895 
0.0898 

0.0878 
o.oSSi 
0.0884 
0.0887 
0.0890 
0.0892 
0.0895 
0.0898 
0.0901 

11.3919 
11-3540 
11.3163 
11.2789 
11.2417 
11.2048 
11.1681 
11.1316 
11.0954 

0.9962 
0.9961 
0.9961 
0.9961 
0.9961 
0.9960 
0.9960 
0.9960 
0.9960 

10 

0.0901 

0.0904 

11.0594 

0.9959 

ii 

12 
13 
M 
15 

16 

17 
18 

19 

20 

0.0903 
0.0906 
0.0909 
0.0912 
0.0915 
0.0918 
0.0921 
0.0924 
0.0927 

0.0907 
0.0910 
0.0913 
0.0916 
0.0919 
0.0922 
0.0925 
0.0928 
0.0931 

11.0237 
10.9882 
10.9529 
10.9178 
10.8829 
10.8483 
10.8139 
10.7797 
10.7457 

0.9959 
0.9959 
0.9959 
0.9958 
0.9958 
0.9958 
0.9958 
0-9957 
0-9957 

0.0929 

0.0934 

10.7119 

0.9957 

21 
22 
23 
24 
25 
26 

27 

28 

29 

30 

3i 
32 
33 
34 

35 
36 

37 
38 
39 
40 

4i 
42 
43 
44 

45 
46 

47 

48 

49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 
00 

0.0932 
0.0935 
0.0938 
0.0941 
0.0944 
0.0947 
0.0950 
0.0953 
0.0956 

0.0936 
0.0939 
0.0942 
0.0945 
0.0948 
0.0951 
0.0954 
0.0957 
0.0960 

10.6783 
10.6450 
10.611  8 
10.5789 
10.5462 
10.5136 
10.4813 
10.4491 
10.4172 

0.9956 
0.9956 
0.9956 
0.9956 
0-9955 
o.9955 
0-9955 
0.9955 
0.9954 

0.0958 

0.0963 

10.385., 

0.9954 

0.0961 
0.0964 
0.0967 
0.0970 
0.0973 
0.0976 
0.0979 
0.0982 
0.0985 

0.0966 
0.0969 
0.0972 
0.0975 
0.0978 
0.0981 
0.0983 
0.0986 
0.0989 

10.3538 
10.3224 
10.2913 
10.2602 
10.229. 
10.1988 
10.1683 
10.1381 
10.1080 

0.9954 
0.9953 
0.9953 
0.9953 
0.9953 
0.9952 

0.9952 
0.9952 
0.9951 

0.0987 

0.0992 

10.0780 

0.9951 

20 
19 

18 

17 

1  6 
15 
14 
13 

12 

1  1 

0.0990 
0.0993 
0.0996 
0.0999 

O.I  OO2 

0.1005 
0.1008 

O.IOII 

0.1013 

0.0995 
0.0998 

O.IOOI 

0.1004 
0.1007 

O.IOIO 

0.1013 
0.1016 
0.1019 

10.0483 
10.0187 
9.9893 
9.9601 
9.9310 
9.9021 

9-8734 
9-8448 
9.8164 

0.9951 
0.9951 
0.9950 
0.9950 
0.9950 
0.9949 
0.9949 
0-9949 
0.9949 

0.1016 

O.IO22 

9.7882 

0.9948 

10 

I 

7 
6 
5 
4 
3 

2 

I 

0 

0.1019 

0.1022 
0.1025 
0.1028 
O.I03I 
0.1034 
0.1037 
0.1039 
Q.f049 

O.I02§ 
0.1028 
0.1030 
0.1033 
O.IO36 
0.1039 
0.1042 
0.1045 
O.IO48 

9.7601 
9.7322 
9.7044 
9.6768 

9-6493 
9.6220 

9-5949 
9.5679 
9.5411 

0.9948 

0.9948 
0.9947 

0.9947 
0.9947 
0.9946 

0.9946 
0.9946 
0.9046 

0-1045 

O.IO51 

9.5144!  o.o'j  J- 

Cos 

Co* 

Tan 

Sin 

' 

264°    *364' 


186°     *276< 


NATURAL 


111 

*97°     187°    *277< 


' 

Sin 

Tan 

Cot 

Cos 

0 

0.1045 

0.1051 

9-5I44 

0.9945 

60 

I 

2 

3 
4 

6 

8 
9 

0.1048 
0.1051 
0.1054 
0.1057 
o.i  060 
0.1063 
0.1066 
0.1068 
0.1071 

0.1054 
0.1057 
o.i  060 
0.1063 
0.1066 
0.1069 
0.1072 
0.1075 
0.1078 

9.4878 
9.4614 
9-4352 
9.4090 
9-383I 
9-3572 

9-33I5 
9.3060 
9.2806 

0-9945 
0.994! 
0.9944 

0.9944 
0.9944 

0-9943 
0.9943 
0.9943 
0.9942 

59 
53 
57 
56 
55 
54 

53 
52 
5i 

10 

0.1074 

0.1080 

9-2553 

0.9942 

50 

ii 

12 
13 
14 

15 

16 

17 

18 

19 

0.1077 
0.1080 
0.1083 
0.1086 
0.1089 
0.1092 
0.1094 
0.1097 

O.IIOO 

0.1083 
0.1086 
0.1089 
0.1092 
0.1095 
0.1098 

O.IIOI 

0.1104 
0.1107 

9.2302 
9.2052 
9.1803 

9-1555 
9.1309 
9.1065 
9.0821 

9-°579 
9-0338 

0.9942 
0.9942 
0.9941 
0.9941 
0.9941 
0.9940 
0.9940 
0.9940 
0.9939 

49 
48 

47 

46 
45 
44 

43 
42 
4i 

20 

0.1103 

O.IIIO 

9.0098 

0.9939 

40 

21  e 
22 
23 
24 
25 
26 

27 
28 
29 

0.1106 
0.1109 

O.III2 

0.1115 
o.mS 

O.II2O 

0.1123 
0.1126 
0.1129 

0.1113 
0.1116 
0.1119 

O.II22 
O.II25 
O.II28 
O.II3I 

O.II33 
O.II36 

8.9860 
8.9623 
8.9387 

8.9152 
8.8919 
8.8686 

8.8455 
8.8225 
8.7996 

0.9939 
0.9938 
0.9938 
0.9938 
0.9937 
0.9937 

o.9937 
0.9936 
0.9936 

39 
-33 
37 
36 
35 
34 

33 
32 
31 

30 

0.1132 

O.II39 

8.7769 

0.0936 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 

0.1135 
0.1138 
0.1141 
0.1144- 

0.1146 

0.1149 
0.1152 

0.1155 

0.1158 

O.II42 
O.II45 
0.1148 
O.II5I 
O.II54 
O.II57 

0.1160 

0.1163 
0.1166 

8.7542 
8.7317 
8.7093 

8.6870 
8.6648 
8.6427 
8.6208 
8.5989 
8-5772 

0.9935 
0-9935 
0-9935 
0-9934 
0.9934 
0-9934 
0-9933 
0-9933 
0.9933 

29 

28 
27 
26 
25 
24 
23 

22 
21 

40 

0.1161 

0.1169 

8-5555 

0.9932 

20 

4i 
42 

43 
44 
45 
46 

47 
43 
49 

0.1164 
0.1167 
0.1170 
0.1172 

0.1175 

0.1178 
0.1181 
0.1184 
0.1187 

0.1172 

O.H75 
0.1178 

o.nSi 
0.1184 
0.1187 
0.1189 
0.1192 
0.1195 

8-5340 
8.5126 
8.4913 
8.4701 
8.4490 
8.4280 
8.4071 
8.3863 
8.3656 

0.9932 
0.9932 
0.9931 
0.9931 
0.9931 
0.9930 
0.9930 
0.9930 
0.9929 

19 

18 
17 
16 
15 
14 
13 

12 
II 

50 

0.1190 

0.1198 

8.3450 

0.9929 

10 

5i 

52 
53 
54 
55 
56 

57 
53 
59 

0.1193 

0.1196 
0.1198 

O.I2OI 

0.1204 
0.1207 

O.I2IO 
O.I2I3 
O.I2I6 

O.I20I 
0.1204 
0.1207 
O.I2IO 
O.I2I3 

0.1216 

O.I2I9 
O.I222 
O.I22§ 

8.324! 
8.3041 
8.2838 
8.2636 
8.2434 
8.2234 
8.2035 
8.1837 
8.1640 

0.9929 
0.9928 
0.9928 
0.9928 
0.9927 
0.9927 
0.9927 
0.9926 
0.9926 

9 

8 

7 
6 
5 
4 

3 

2 
I 

60 

O.I2I9 

0.1228 

8.1443 

0.9925 

0 

Cos 

Cot 

Tan 

Sin 

' 

173°     263°     *353°  83° 


NATURAL 


' 

Sin 

Tan 

Cot 

Cos 

0 

O.I2IQ 

0.1228 

8.1443 

0.9925 

60 

59 

53 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 

47 
46 
45 
44 

43 
42 
4i 
40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 

27 
26 
25 
24 
23 

22 
21 
20 
19 

1  8 
17 
16 

15 
14 
13 

12 
II 

I 

2 

3 
4 

6 

8 
9 

O.I222 
0.1224 
0.1227 
O.I23O 
0.1233 
0.1236 
0.1239 
0.1242 
0.1245 

0.1231 
0.1234 
0.1237 
0.1240 
0.1243 
0.1246 
0.1249 
0.1251 
0.1254 

8.1248 
8.1054 
8.0860 
8.0667 
8.0476 
8.0285 
8.0095 
7.9906 
7.9718 

0.9925 
0.9925 
0.9924 
0.9924 
0.9924 
0.9923 
0.9923 
0.9923 
0.9922 

10 
ii 

12 
13 
14 
15 

16 

17 

18 

J9 

20 

21 

22 
23 
24 

25 
26 

27 
28 
29 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 
42 

43 
44 
45 
46 
47 
48 
49 
50 

5i 
52 
53 
54 
55 
56 

57 
53 
59 
60 

0.1248 

0.1257 

7-9530 

0.9922 

0.1250 
0.1253 
0.1256 
0.1259 
0.1262 
0.1265 
0.1268 
O.I27I 
0.12/4 

0.1260 
0.1263 
0.1266 
0.1269 
0.1272 
0.1275 
0.1278 
0.1281 
0.1284 

7-9344 
7-9158 
7.8973 
7.8789 
7.8606 
7.8424 
7.8243 
7.8062 
7.7882 

0.9922 
0.9921 
0.9921 
0.9920 
0.9920 
0.9920 
0.9919 
0.9919 
0.9919 

O.I2/6 

0.1287 

7.7704 

0.9918 

0.1279 
0.1282 
0.1285 
0.1288 
0.129! 
0.1294 
0.1297 
O.I299 
O.I3O2 

0.1290 
0.1293 
0.1296 
0.1299 
0.1302 
0.1305 
0.1308 
0.1311 
0.1314 

7.7525 
7.7348 
7.7171 

7.6996 
7.6821 
7.6647 

7.6473 
7.6301 
7.6129 

0.9918 
0.9917 
0.9917 
0.9917 
0.9916 
0.9916 
0.9916 
0.9915 
0.9915 

0.1305 

0.1317 

7.5958 

0.9914 

0.1308 
O.I3II 
O.I3I4 
O.I3I7 
O.I32O 
0.1323 

0.1325 
0.1328 
O.I33I 

0.1319 
0.1322 

0.1325 
0.1328 
0.1331 
o.i334 
0.1337 
0.1340 
0.1343 

7.5787 
7.5618 

7-5449 
7.5281 
7.5H3 
7-4947 
7.4781 
74615 
74451 

0.9914 
0.9914 
0.9913 
0.9913 
0.9913 
0.9912 
0.9912 
0.9911 
0.9911 

0.1334 

0.1346 

7.4287 

0.9911 

0.1337 
0.1340 

0.1343 
0.1346 
0.1349 
O.I35I 
0.1354 
0.1357 
0.1360 

0.1349 
0.1352 
o.i355 
0.1358 
0.1361 
0.1364 

0.1367 
0.1370 

0.1373 

7.4124 
7.3962 
7.3800 

7.3639 
7-3479 
7.3319 
7.3160 
7.3002 
7.2844 

0.9910 
0.9910 
0.9909 
0.9909 
0.9909 
0.9908 
0.9908 
0.9907 
0.9907 

0.1363 

0.1376 

7.2687 

0.9907 

10 

0.1366 
0.1369 
0.1372 

0.1374 
0.1377 
0.1380 

0.1383 
0.1386 
0.1389 

0.1379 
0.1382 
0.1385 
0.1388 

0.1391 
0.1394 

0.1397 

0.1399 
0.1402 

7.2531 
7.2375 
7.2220 

7.2066 
7.1912 
7-1759 
7.1607 

7.1455 
7.1304 

0.9906 
0.9906 
0.9905 
0.9905 
0.9905 
0.9904 
0.9904 
0.9903 
0.9903 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.1392 

0.1405 

7.H54 

0.9903 

Cos 

Cot 

Tan 

Sin 

' 

262°     *352' 


112 

*98< 


188°  *278° 


8° 


NATURAL 


9°      *99°    189°  *279° 


' 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 
5 
6 

7 
8 

9 
10 
ii 

12 
13 
14 

15 

16 

17 

18 

19 

20 

21 
22 
23 
24 

25 
26 

27 
28 

29 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 

42 
43 
44 
45 
46 

47 
48 
49 
50 

0.1392 

0.1405 

7-II54 

0.9903 

60 

59 
58 
57 
56 
55 
54 
53 
52 
5i 

0.1395 
0.1397 
0.1400 

0.1403 
0.1406 
0.1409 
0.1412 
0.1415 
0.1418 

0.1408 
0.1411 
0.1414 
0.1417 
0.1420 
0.1423 
0.1426 
0.1429 
0.1432 

7.1004 

7.0855 
7.0706 

7.0558 
7.0410 
7.0264 
7.0117 
6.9972 
6.9827 

0.9902 
0.9902 
0.9901 
0.9901 
0.9901 
0.9900 
0.9900 
0.9899 
0.9899 

0.1421 

0.1435 

6.9682 

0.9899 

50 

49 

48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 

0.1423 
0.1426 
0.1429 

0.1432 

0.143! 
0.1438 

0.1441 
0.1444 
0.1446 

0.1438 
0.1441 
0.1444 

o.i447 
0.1450 

o.i453 
0.1456 
0.1459 
0.1462 

6.9538 
6-9395 
6.9252 

6.9110 
6.8969 
6.8828 
6.8687 
6.8548 
6.8408 

0.9898 
0.9898 
0.9897 
0.9897 
0.9897 
0.9896 
0.9896 
0.9895 
0.9895 

0.1449 

0.1465 

6.8269 

0.9894 

0.1452 

o.i455 
0.1458 

0.1461 
0.1464 
0.1467 
0.1469 
0.1472 
0.1475 

0.1468 
0.1471 
o.i474 
0.1477 
0.1480 
0.1483 
0.1486 
0.1489 
0.1492 

6.8131 

6-7994 
6.7856 

6.7720 
6.7584 
6.7448 

6.7313 
6.7179 
6.7045 

0.9894 
0.9894 
0.9893 
0.9893 
0.9892 
0.9892 
0.9891 
0.9891 
0.9891 

0.1478 

0.1495 

6.6912 

0.9890 

30 

29 
28 
27 
26 
25 
24 
23 

22 
21 
20 
I9 

18 
17 
16 
15 
14 
13 

12 
II 
10 

9 

8 

7 
6 
5 
4 
3 

2 
I 

0 

0.1481 
0.1484 
0.1487 
0.1490 
0.1492 
0.1495 
0.1498 
0.1501 
0.1504 

0.1497 
0.1500 
0.1503 
0.1506 
0.1509 
0.1512 

0.1515 
0.1518 
0.1521 

6.6779 
6.6646 
6.6514 
6.6383 
6.6252 
6.6122 
6.5992 
6.5863 
6-5734 

0.9890 
0.9889 
0.9889 
0.9888 
0.9888 
0.9888 
0.9887 
0.9887 
0.9886 

0.1507 

0.1524 

6.5606 

0.9886 

0.1510 

0.1513 
0.1515 

0.1518 
0.1521 
0.1524 
0.1527 
0.1530 
0.1533 

0.1527 
0.1530 
0.1533 
0.1536 

0.1539 
0.1542 

o.i545 
0.1548 

0.1551 

6.5478 
6-5350 
6.5223 

6.5097 
6.4971 
6.4846 
6.4721 
6.4596 
6.4472 

0.9885 
0.9885 
0.9884 
0.9884 
0.9884 
0.9883 
0.9883 
0.9882 
0.9882 

0.1536 

o.i554 

6.4348 

0.9881 

5i 
52 
53 
54 
55 
56 

57 
58 
59 

0.1538 
0.1541 
0.1544 

0.1547 
0.1550 
0.1553 
0.1556 

0.1559 
0.1561 

o.i557 
0.1560 
0.1563 
0.1566 
0.1569 
0.1572 

0.157! 
0.1578 
0.1581 

6.4225 
6.4103 
6.3980 

6.3859 
6.3737 
6.3617 

6.3496 
6.3376 
6.3257 

0.9881 
0.9880 
0.9880 
0.9880 
0.9879 
0.9879 
0.9878 
0.9878 
0.9877 

HO 

0.1564 

0.1584 

6.3138 

0.9877 

Cos 

Cot 

Tan 

Sin 

' 

' 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 

6 

7 
8 

9 
10 

ii 

12 
13 
14 
15 

16 

17 

18 

*9 

20 

21 
22 
23 

24 
25 
26 

27 

28 

29 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 

42 
43 
44 
45 
46 

47 
48 
49 
50 

5i 

52 
53 
54 
55 
56 

57 
58 
59 
00 

0.1564 

0.1584 

6.3138 

0.9877 

60 

0.1567 
0.1570 
o.i573 
0.1576 

o.i579 
0.1582 

0.1584 
0.1587 
0.1590 

0.1587 
0.1590 
o.i593 
0.1596 
0.1599 
0.1602 
0.1605 
o.i  608 
0.1611 

6.3019 
6.2901 
6.2783 
6.2666 
6.2549 
6.2432 
6.2316 

6.  2  20O 

6.2085 

0.9876 
0.9876 
0.9876 
0.9875 
0.9875 
0.9874 
0.9874 
0.9873 
0.9873 

59 
58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 

25 
24 

23. 

22 
21 
20 
19 

18 
17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

o.i593 

0.1614 

6.1970 

0.9872 

0.1596 

o.i599 
0.1602 

0.1605 
0.1607 
0.1610 
0.1613 
0.1616 
0.1619 

0.1617 
0.1620 
0.1623 
0.1626 
0.1629 
0.1632 
0.1635 
0.1638 
0.1641 

6.1856 
6.1742 
6.1628 

6.1515 

6.1402 
6.1290 
6.1178 
6.1066 

6.0955 

0.9872 
0.9871 
0.9871 
0.9870 
0.9870 
0.9869 
0.9869 
0.9869 
•0.9868 

0.1622 

0.1644 

6.0844 

0.9868 

0.1625 
0.1628 
0.1630 
0.1633 
0.1636 
0.1639 
0.1642 
0.1645 
0.1648 

0.1647 
0.1650 
0.1653 
0.1655 
0.1658 
0.1661 
0.1664 
0.1667 
0.1670 

6.0734 

6.0624 
6.0514 
6.0405 
6.0296 
6.0188 
6.0080 

5.9972 
5.9865 

0.9867 
0.9867 
0.9866 
0.9866 
0.9865 
0.9865 
0.9864 
0.9864 
0.9863 

0.1650 

0.1673 

5-9758 

0.9863 

0.1653 
0.1656 
0.1659 
0.1662 
0.1665 
0.1668 
0.1671 
0.1673 
0.1676 

0.1676 
0.1679 
0.1682 
0.1685 
0.1688 
0.1691 
0.1694 
0.1697 
0.1700 

5.9651 

5-9545 
5-9439 

5-9333 
5.9228 
5.9124 
5.9019 

5-8915 
5.8811 

0.9862 
0.9862 
0.9861 
0.9861 
0.9860 
0.9860 
0.9859 
0.9859 
0.9859 

0.1679 

0.1703 

5-8708 

0.9858 

0.1682 
0.1685 
0.1688 
0.1691 
0.1693 
0.1696 
0.1699 
0.1702 
0.1705 

0.1706 
0.1769 
0.1712 
0.1715 
0.1718 
0.1721 
0.1724 
0.1727 
0.1730 

5-8605 
5-8502 
5.8400 
5.8298 
5.8197 
5-8095 

5.7994 
5-7894 
5-7794 

0.9858 
0.9857 
0.9857 
0.9856 
0.9856 
0.9855 

0.9855 
0.9854 
0.9854 

0.1708 

0.1733 

5.7694 

0.9853 

0.1711 
0.1714 
0.1716 
0.1719 
0.1722 
0.1725 
0.1728 
0.1731 
0.1734 

0.1736 
0.1739 
0.1742 

0.1745 
0.1748 
0.1751 

0.1754 
0.1757 
0.1760 

5-7594 
5-7495 
5.7396 
5.7297 
5.7I99 
5.7101 

5-7004 
5.6906 
5.6809 

0.9853 
0.9852 
0.9852 
0.9851 
0.9851 
0.9850 
0.9850 
0.9849 
0.9849 

0.1736  0.1763 

5.6713  |  0.98.18 

Cos 

Cot 

Tan 

Sin 

261°'*351° 


NATURAL 


260°  *350< 


*100°  190°  *280° 


NATURAL 


ir 


*101 


113 
191°  *281° 


/ 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 
5 
6 

7 
8 

9 
10 
ii 

12 
13 
14 

15 

16 

17 

18 

!9 

20 

21 
22 
23 
24 
25 
26 

27 

28 

29 

30 

3i 
32 
33 
34 

35 
36 

37 

38 
39 
40 

4i 
42 

43 
44 
45 
46 

47 

48 

49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 
60 

0.1736 

0.1763 

5.6713 

0.9848 

60 

59 

58 
57 
56 
55 
54 
53 
52 
5i 
50 

49 

48 

47 
46 
45 
44 

43 
42 

4i 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 

28 

27 
26 
25 
24 
23 

22 
21 

20 

19 

18 

17 
16 
15 
14 
13 

12 
II 

10 

9 
8 

7 
6 
5 
4 
3 

2 
I 
0 

o.i739 
0.1742 

o.i745 
0.1748 
0.1751 
o.i754 
o.i757 
o.i759 
0.1762 

0.1766 
0.1769 
0.1772 

o.i775 
0.1778 
0.1781 
0.1784 
0.1787 
0.1790 

5.6617 
5-6521 
5-6425 

5-6329 
5-6234 
5.6140 

5-6045 
5-5951 
5-5857 

0.9848 
0.9847 
0.9847 
0.9846 
0.9846 
0.9845 
0.9845 
0.9844 
0.9843 

0.1765 
0.1768 
0.1771 
0.1774 
0.1777 
0.1779 
0.1782 

0.1785 
0.1788 
0.1791 

0-1793 

5-5764 

0.9843 

0.1796 

0.1799 
0.1802 

0.1805 
o.i  808 
0.1811 
0.1814 
0.1817 
0.1820 

5-5671 
5.5578 
5.5485 

5.5393 
5.5301 
5.5209 

5-5118 
5-5026 
54936 

0.9842 
0.9842 
0.9841 
0.9841 
0.9840 
0.9840 
0.9839 
0.9839 
0.9838 

0.1794 

0.1823 

5-4845 

0.9838 

0.1797 
0.1799 
0.1802 
0.1805 
0.1808 
0.1811 
0.1814 
0.1817 
0.1819 

0.1826 
0.1829 
0.1832 
0.1835 
0.1838 
0.1841 
0.1844 
0.1847 
0.1850 

5-4755 
5-4665 
5-4575 
54486 
5-4397 
54308 

54219 
5-4131 
54043 

0.9837 
0.9837 
0.9836 
0.9836 
0.9835 
0-9835 
0.9834 
0.9834 
0.9833 

0.1822 

0.1853 

5-3955 

0.9833 

0.1825 
0.1828 
0.1831 
0.1834 
0.1837 
0.1840 
0.1842 
0.1845 
0.1848 

0.1856 
0.1859 
0.1862 
0.1865 
0.1868 
0.1871 
0.1874 
0.1877 
0.1880 

5.3868 
5.3781 
5.3694 
5o6o7 
5.3521 
5-3435 

5-3349 
5-3263 
5-3178 

0.9832 
0.9831 
0.9831 
0.9830 
0.9830 
0.9829 
0.9829 
0.9828 
0.9828 

0.1851 

0.1883 

5-3093 

0.9827 

0.1854 
0.1857 

O.I  860 

0.1862 

0.1865 
0.1868 
0.1871 

0.1874 
0.1877 

0.1887 
0.1890 
0.1893 
0.1896 
0.1899 
0.1902 
0.1905 
0.1908 
0.1911 

5-3008 
5.2924 
5-2839 

5-2755 
5.2672 
5-2588 

5.2505 
5.2422 

5-2339 

0.9827 
0.9826 
0.9826 
0.9825 
0.9825 
0.9824 
0.9823 
0.9823 
0.9822 

0.1880 

0.1914 

5-2257 

0.9822 

0.1882 
0.1885 
0.1888 
0.1891 
0.1894 
0.1897 
0.1900 
0.1902 
0.1905 

0.1917 
0.1920 
0.1923 
0.1926 
0.1929 
0.1932 

0.1935 
0.1938 
0.1941 

5-2174 
5.2092 
5.2011 
5.1929 
5-1848 
5.1767 
5.1686 
5.1606 
5-1526 

0.9821 
0.9821 
0.9820 
0.9820 
0.9819 
0.9818 
0.9818 
0.9817 
0.9817 

0.1908 

0.1944 

5.1446 

0.9816 

Cos 

Cot 

Tan 

Sin 

' 

*169°  259°  *349°      79' 


NATURAL 


' 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 
4 
5 
6 

1 
8 

9 
10 
ii 

12 
13 
14 
15 

16 

17 

18 

19 
20 

21 

22 
23 
24 

25 
26 

27 

28 

29 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 

42 

43 
44 

45 
46 

47 

48 

49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 
60 

0.1908 

0.1944 

5.1446 

0.9816 

60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 

48 
47 
46 
45 
44 

43 

42 

4i 
40 

39 
38 
3V 
36 
35 
34 

33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 

22 
21 

20 

19 

18 
17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 

5 
4 
3 

2 

I 

0 

0.1911 
0.1914 
0.1917 
0.1920 
0.1922 
0.1925 
0.1928 
0.1931 
0.1934 

0.1947 
0.1950 
0.1953 
0.1956 
0.1959 
0.1962 
0.1965 
0.1968 
0.1971 

5-1366 
5.1286 
5.1207 
5.1128 
5.1049 
5.0970 
5.0892 
5.0814 
5-0736 

0.9816 
0.9815 
0.9815 
0.9814 
0.9813 
0.9813 
0.9812 
0.9812 
0.9811 

0.1937 

0.1974 

5.0658 

0.9811 

0.1939 
0.1942 
0.1945 
0.1948 
0.1951 
0.1954 

0.1957 
0.1959 
0.1962 

0.1977 
0.1980 
0.1983 
0.1986 
0.1989 
0.1992 
0.1995 
0.1998 

O.2OOI 

5-0581 
5-0504 
5.0427 

5-0350 
5-0273 
5-0197 
5.0121 
5-0045 
4.9969 

0.9810 
0.9810 
0.9809 
0.9808 
0.9808 
0.9807 
0.9807 
0.9806 
0.9806 

0.1965 

O.2OO4 

4.9894 

0.9805 

0.1968 
0.1971 
0.1974 
0.1977 

0.1979 
0.1982 

0.1985 
0.1988 
0.1991 

O.2OO7 
O.2OIO 
0.2013 
O.2OI6 
O.2OI9 
O.2O22 
0.2025 
O.2O28 
O.2O3I 

4.9819 

4-9744 
4.9669 

4-9594 
4.9520 
4.9446 

4-9372 
4.9298 
4.9225 

0.9804 
0.9804 
0.9803 
0.9803 
0.9802 
0.9802 
0.9801 
0.9800 
0.9800 

0.1994 

0.2035 

4-9*52 

0.9799 

0.1997 
0.1999 

0.2002 
0.2005 
0.2008 
0.201  1 
O.2OI4 

0.2016 

O.2OI9 

0.2038 
O.2O4I 
O.2O44 
O.2O47 
O.2O5O 
0.2053 
O.2O56 
O.2O59 
O.2O62 

4.9078 
4.9006 
4.8933 
4.8860 
4.8788 
4.8716 
4.8644 

4.8573 
4.8501 

0-9799 
0.9798 
0.9798 
0.9797 
0.9796 
0.9796 

0.9795 
0-9795 
0.9794 

0.2022 

O.2O65 

4.8430 

0.9793 

0.2025 
O.2O28 
O.2O3I 
0.2034 
O.2O36 
0.2039 
0.2042 
O.2O45 
O.2O48 

O.2O68 
O.2O7I 
0.2074 
O.2O77 
O.2O8O 
0.2083 
0.2086 
0.2089 
O.2O92 

4.8359 
4.8288 
4.8218 
4.8147 
4.8077 
4.8007 

4.7937 
4.7867 
4J798 

0.9793 
0.9792 
0.9792 
0.9791 
0.9790 
0.9790 
0.9789 
0.9789 
0.9788 

0.2051 

O.2O95 

4.7729 

0.9787 

0.2054 
0.2056 
0.2059 
0.2062 
0.2065 
O.2O68 
0.2071 
0.2073 
O.2O76 

0.2098 
O.2IOI 
0.2104 
O.2IO7 
0.2IIO 
0.2II3 
0.2II6 
0.2II9 
O.2I23 

4-7659 
4-7591 
4.7522 

4.7453 
4.7385 
4.7317 
4.7249 
4.7181 
4.7114 

0.9787 
0.9786 
0.9786 
0.9785 
0.9784 
0.9784 
0.9783 
0.9783 
0.9782 

O.2O79 

O.2I26 

4.7046 

0.9781 

Cos  1  Cot 

Tan 

Sin 

' 

78' 


*168°  258°  *348< 


114 

*102°  192°  *282° 


NATURAL 


13C 


*103°  193°  *283° 


' 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 
5 
6 

8 
9 
10 
ii 

12 
13 
14 

15 

16 

17 

18 

19 
20 

21 

22 
23 
24 
25 
26 

27 
28 
29 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 
42 

43 
44 
45 
46 

47 
48 

49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 
60 

0.2079 

0.2126 

4.7046 

0.9781 

60 

59 

58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 

28 

271 
26 

25 
24 
23 

22 
21 

20 

19 

18 
17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 
I 

0 

0.2082 
0.2085 
0.2088 
0.2090 
0.2093 
0.2096 
0.2099 

0.2102 
O.2IO5 

0.2129 
0.2132 
0.2135 
0.2138 
0.2141 
0.2144 
0.2147 
0.2150 
0.2153 

4.6979 
4.6912 
4.6845 
4.6779 
4.6712 
4.6646 
4.6580 
4.6514 
4.6448 

0.9781 
0.9780 
0.9780 

0.9779 
0.9778 

0.9778 
0.9777 
0.9777 
0.9776 

0.2108 

0.2156 

4.6382 

0-9775 

O.2IIO 
0.2II3 

0.2116 

0.2119 
O.2I22 
O.2I25 
O.2I27 
O.2I3O 
0.2133 

0.2159 
0.2162 
0.2165 
0.2168 
0.2171 
0.2174 
0.2177 
0.2180 
o!2i83 

4-6317 
4.6252 
4.6187 
4.6122 
4.6057 
4-5993 
4.5928 
4.5864 
4.5800 

0-9775 
o.9774 
o.9774 

o.9773 
0.9772 
0.9772 

0.9771 
0.9770 
0.9770 

0.2136 

0.2186 

4.5736 

0.9769 

0.2139 
O.2I42 
0.2145 
0.2147 
0.2150 
0.2153 
O.2I56 
0.2159 
0.2162 

0.2189 
0.2193 
0.2196 
0.2199 

O.22O2 
0.220§ 
0.2208 
0.2211 
O.22I4 

4.5673 
4-5609 
4.5546 

4.5483 
4.5420 

4-5357 
4.5294 
4-5232 
4-5169 

0.9769 
0.9768 
0.9767 

0.9767 
0.9766 
0.9765 
0.9765 
0.9764 
0.9764 

0.2164 

O.22I7 

4-5107 

0.9763 

0.2167 
O.2I7O 
0.2173 
O.2I76 
0.2179 
0.21  8l 

0.2184 
0.2187 
0.2190 

O.222O 
O.2223 
0.2226 
O.2229 
0.2232 
0.2235 
0.2238 
0.2241 
O.2244 

4.5045 
4.4983 
4.4922 
4.4860 
4-4799 
4-4737 
4.4676 
44615 
4-4555 

0.9762 
0.9762 
0.9761 
0.9760 
0.9760 
o.9759 

0.9759 
0.9758 
0-9757 

0.2193 

O.2247 

4-4494 

0-9757 

0.2196 
0.2198 

0.2201 
0.2204 
0.2207 
O.22IQ 
0.2213 
O.22I5 
O.22I8 

0.2251 
0.2254 
0.2257 
O.226O 
0.2263 
O.2266 
0.2269 
O.2272 
0.2275 

4-4434 
4-4373 
4-4313 

44253 
4.4194 

4-4I34 

4-4075 
4.4015 
4-3956 

0.9756 
0-9755 
0-9755 
o.9754 
0.9753 
0-9753 
0.9752 
0.9751 
0.9751 

O.222I 

0.2278 

4-3897 

0.9750 

0.2224 
O.2227 
O.223O 
0.2233 
0.2235 
0.2238 
O.224I 
0.2244 
0.2247 

O.228I 
O.2284 
0.2287 
O.229O 
0.2293 
0.2296 
0.2299 
0.2303 
0.2306 

4-3838 
4-3779 
4-3721 
4.3662 
4-3604 
4.3546 
4.3488 
4-3430 
4-3372 

.0.9750 
0.9749 
0.9748 
0.9748 
0.9747 
0.9746 
0.9746 
0-9745 
0-9744 

0.225O   0.2309 

4-33I5 

0.9744 

Cos 

Cot 

Tan  |  Sin 

' 

257°  *347° 


NATURAL 


' 

Sin 

Tan 

Cot  |  Cos 

0 
I 

2 

3 

4 
5 
6 

8 
9 
10 
ii 

12 
13 
14 
15 

16 

17 

18 

19 
20 

21 
22 
23 
24 

25 
26 

27 
28 
29 

30 

3i 
32 
33 
34 

35 
36 

37 
38 
39 
40 

4i 

42 

43 
44 
45 
46 

47 
48 
49 
50 

5i 

52 
53 
54 
55 
56 

57 
58 
59 
60 

0.2250 

0.2309 

4-3315 

0.9744 

60 

59 

58 
57 
56 
55 
54 
53 
52 
5i 
50 

49 
48 
47 
46 
45 
44 

43 
42 

4i 
40 

39 

38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 
28 
27 
26 

25 
24 

23 

22 
21 

20 

19 

18 

17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.2252 
0.2255 
0.2258 
0.2261 
0.2264 
0.2267 
0.2269 
0.2272 
0.2275 

0.2312 
0.2315 
0.2318 
0.2321 
0.2324 
0.2327 
0.2330 
0.2333 
0.2336 

4-3257 
4.3200 

4-3143 
4.3086 
4.3029 
4.2972 
4.2916 
4-2859 
4.2803 

0.9743 
0.9742 
0.9742 
0.9741 
0.9740 
0.9740 

0.9739 
0.9738 
0.9738 

0.2278 

0.2339 

4.2747 

0.9737 

0.2281 
0.2284 
0.2286 
0.2289 
0.2292 
0.2295 
0.2298 
0.2300 
0.2303 

0.2342 

0.2345 
0.2349 

0.2352 
0.2355 
0.2358 
0.2361 
0.2364 
0.2367 

4.2691 
4-2635 
4-2580 

4-2524 
4.2468 
4.2413 

4.2358 
4.2303 
4.2248 

0.9736 
0.9736 
0.9735 
0.9734 
0.9734 
0.9733 
0.9732 
0.9732 
0.9731 

0.2306 

0.2370 

4.2193 

0.9730 

0.2309 
0.2312 
0.2315 
0.2317 
0.2320 
0.2323 
0.2326 
0.2329 
0.2332 

0.2373 
0.2376 
0.2379 
0.2382 
0.2385 
0.2388 
0.2392 

0.2395 
0.2398 

4.2139 
4.2084 
4.2030 
4.1976 
4.1922 
4.1868 
4.1814 
4.1760 
4.1706 

0.9730 
0.9729 
0.9728 
0.9728 

0.9727 
0.9726 

0.9726 
0.9725 
0.9/24 

°-2334 

0.2401 

4-1653 

0.9-24 

0.2337 
0.2340 
0.2343 
0.2346 
0.2349 
0.2351 

0.2354 
0.2357 
0.2360 

0.2404 
0.2407 
0.2410 
0.2413 
0.2416 
0.2419 
0.2422 
0.2425 
0.2428 

4.1600 
4.1547 
4-1493 
4.1441 
4.1388 
4.1335 
4.1282 
4.1230 
4.1178 

0.9723 
0.9722 
0.9722 
0.9721 
0.9720 
0.9720 
0.9719 
0.9718 
0.9718 

0.2363 

0.2432 

4.1126 

0.9717 

0.2366 
0.2368 
0.2371 
0.2374 

0.2377 
0.2380 

0.2383 
0.2385 
0.2388 

0.2435 
0.2438 
0.2441 
0.2444 
0.2447 
0.2450 

0.2453 
0.2456 
0.2459 

4.1074 
4.1022 
4.0970 
4.0918 
4.0867 
4-0815 
4.0764 

4-0713 
4.0662 

0.9716 
0.9715 
0.9715 
0.9714 
0.9713 
0.9713 
0.9712 
0.9711 
0.9711 

0.2391 

0.2462 

4.0611 

0.9710 

0.2394 
0.2397 
0.2399 
0.2402 
0.2405 
0.2408 
0.2411 
0.2414 
0.2416 

0.2465 
0.2469 
0.2472 
0.2475 
0.2478 
0.2481 

0.2484 
0.2487 
0.2490 

4.0560 
4.0509 
4-0459 
4.0408 
4.0358 
4.0308 
4.0257 
4.0207 
4-0158 

0.9709 
0.9709 
0.9708 

0.9707 
0.9706 
0.9706 
0.9705 
0.9704 
0.9704 

0.2419   0.2493 

4.0108 

0.9703 

Cos 

Cot 

Tan 

Sin 

' 

76' 


*166°  256°  *346° 


*104°  194°  *284 


NATURAL 


15°    *105°     195C 


115 

*285° 


Sin 

Tan 

Cot 

Cos 

0 

0.2419 

0.2493 

4.0108 

0.9703 

60 

I 

2 

3 
4 
5 
6 

7 

8 

9 

0.2422 
0.2425 
0.2428 
0.2431 
0.2433 
0.2436 
0.2439 
0.2442 
0.2445 

0.2496 
0.2499 
0.2503 
0.2506 
0.2509 
0.2512 
0.2515 
0.2518 
0.2521 

4.0058 
4.0009 
3-9959 
3.9910 
3.9861 
3-9812 

3-9763 
3.9714 
3-9665 

0.9702 
0.9702 
0.9701 
0.9700 
0.9699 
0.9699 
0.9698 
0.9697 
0.9697 

59 

58 
57 
56 
55 
54 

53 
52 
5i 

10 

0.2447 

0.2524 

3-9617 

0.9696 

50 

II 

12 
13 
14 

15 
16 

17 

18 
19 

0.2450 

0.2453 
0.2456 

0.2459 
0.2462 
0.2464 
0.2467 
0.2470 
0.2473 

0.2527 
0.2530 
0.2533 
0.2537 
0.2540 
0.2543 
0.2546 
0.2549 
0.2552 

3.9568 
3.9520 
3-9471 
3.9423 
3-9375 
3.9327 
3.9279 
3-9232 
3.9184 

0.9695 
0.9694 
0.9694 
0.9693 
0.9692 
0.9692 
0.9691 
0.9690 
0.9689 

49 
48 
47 
46 
45 
44 

43 
42 

4i 

20 

0.2476 

0.2555 

3-9136 

0.9689 

40 

21 
22 
23 
24 

25 
26 

27 
28 
2Q 

0.2478 
0.2481 
0.2484 
0.2487 
0.2490 
0.2493 
0.2495 
0.2498 
0.2501 

0.2558 
0.2561 
0.2564 
0.2568 
0.2571 
0.2574 

0.2577 
0.2580 
0.2583 

3-9089 
3.9042 
3.8995 

3.8947 
3.8900 

3.8854 
3.8807 
3.8760 
3.S/I4 

0.9688 
0.9687 
0.9687 
0.9686 
0.9685 
0.9684 
0.9684 
0.9683 
0.9682 

39 

38 
37 
36 
35 
34 

33 
32 
3i 

30 

0.2504 

0.2586 

3.8007 

0.9681 

30 

31 

32 

33 
34 
35 
36 

37 
38 
39 

0.2507 
0.2509 
0.2512 
0.2515 
0.2518 
0.2521 
0.2524 
0.2526 
0.2529 

0.2589 
0.2592 
0.2595 
0.2599 
0.2602 
0.2605 
0.2608 
0.2611 
0.2614 

3.8621 

3.8575 
3-8528 

3.8482 
3-8436 
3.8391 

3.8345 
3-8299 

3-8254 

0.9681 
0.9680 
0.9679 
0.9679 
0.9678 
0.9677 
0.9676 
0.9676 
0.9675 

29 

28 

27 
26 
25 
24 
23 

22 
21 

40 

0.2532 

0.2617 

3.8208 

0.9674 

20 

4i 

.  42 
43 
44 
45 
46 

47 

48 
49 

0.2535 
0.2538 
0.2540 

0.2543 
0.2546 
0.2549 
0.2552 
0-2554 
0.2557 

0.2620 
0.2623 
0.2627 
0.2630 
0.2633 
0.2636 
0.2639 
0.2642 
0.2645 

3.8163 
3.8118 
3-8073 
3.8028 
3-7983 
3.7938 

3-7893 
3.7848 
3-7804 

0.9673 
0.9673 
0.9672 
0.9671 
0.9670 
0.9670 
0.9669 
0.9668 
0.9667 

19 

18 
17 
16 
15 
M 
13 

ir 

50 

0.2560 

0.2648 

3.7760 

0.9667 

10 

5i 

52 
53 
54 
55 
56 

57 
58 
59 

0.2563 
0.2566 
0.2569 
0.2571 
0.2574 
0.2577 
0.2580 
0.2583 
0.2585 

0.2651 
0.2655 
0.2658 
0.2661 
0.2664 
0.2667 
0.2670 
0.2673 
0.2676 

3.7715 
3-7671 
3.7627 

3.7583 
3.7539 
3-7495 

3-7451 
3.7408 

3.7364 

0.9666 
0.9665 
0.9665 
0.9664 
0.9663 
0.9662 
0.9662 
0.9661 
0.9660 

9 

8 

7 
6 
5 
4 
3 

2 
I 

00 

0.2588 

0.2679 

3-7321 

0.9659 

0 

Cos 

Cot 

Tan 

Sin 

' 

; 

Sin 

Tan 

Cot 

Cos 

0 

0.2588 

0.2679 

3-7321 

0.9659 

60 

I 

2 

3 
4 
5 
6 

8 
9 

0.2591 
0.2594 
0.2597 
0.2599 
0.2602 
0.2605 
0.2608 
0.2611 
0.2613 

0.2683 
0.2686 
0.2689 
0.2692 
0.2695 
0.2698 
0.2701 
0.2704 
0.2708 

3-7277 
3-7234 
3-7I91 
3-7148 
3-7105 
3.7062 

3.7019 
3.6976 

3-6933 

0.9659 
0.9658 
0.9657 
0.9656 
0.9655 
0.965! 
0.9654 

0.9653 
0.9652 

59 

58 
57 
56 
55 
54 

53 
52 
5i 

10 

0.2616 

0.2711 

3.6891 

0.9652 

50 

ii 

12 
13 
14 

15 

16 

17 

18 

19 

0.2619 
0.2622 
0.2625 
0.2628 
0.2630 
0.2633 
0.2636 
0.2639 
0.2642 

0.2714 
0.2717 
0.2720 
0.2723 
0.2726 
0.2729 

0.2733 
0.2736 
0.2739 

3.6848 
3.6806 
3.6764 
3.6722 
3.6680 
3.6638 

3-6596 
3-6554. 
3-6512 

0.9651 
0.9650 
0.9649 
0.9649 
0.9648 
0.9647 
0.9646 
0.9646 
0.9645 

49 

48 

47 
46 
45 
44 

43 
42 

41 

20 

0.2644 

0.2742 

3.6470 

0.9644 

40 

21 

22 
23 
24 

25 
26 

27 
28 
29 

0.2647 
0.2650 
0.2653 
0.2656 
0.2658 
0.2661 
0.2664 
0.2667 
0.2670 

0.2745 
0.2748 
0.2751 
0.2754 
0.2758 
0.2761 
0.2764 
0.2767 
0.2770 

3-6429 

3-6387 
3-6346 

3-6305 
3.6264 
3.6222 
3.6181 
3.6140 
3.6100 

0.9643 
0.9642 
0.9642 
0.9641 
0.9640 
0.9639 
0.9639 
0.9638 
0.9637 

39 
38 
37 
36 
35 
34 
33 
32 
31 

30 

0.2672 

0.2773 

3-6059 

0.0636 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 

0.2675 
0.2678 
0.2681 
0.2684 
0.2686 
0.2689 
0.2692 
0.2695 
0.2698 

0.2776 
0.2780 
0.2783 
0.2786 
0.2789 
0.2792 
0.2795 
0.2798 
0.2801 

3.6018 
3.5978 
3-5937 

3.5897 
3-5856 
3-5816 

3.5776 
3.5736 
3-5696 

0.9636 

0.963! 
0.9634 

0.9633 
0.9632 
0.9632 
0.9631 
0.9630 
0.9629 

29 
28 
27 
26 
25 
24 
23 

22 
21 

40 

0.2700 

0.2805 

3-5656 

0.9628 

20 

4i 

42 
43 
44 
45 
46 

47 
48 

49 

0.2703 
0.2706 
0.2709 
0.2712 
0.2714 
0.2717 
0.2720 
0.2723 
0.2726 

0.2808 
0.2811 
0.2814 
0.2817 
0.2820 
0.2823 
0.2827 
0.2830 
0.2833 

3-5616 
3.5576 
3.5536 

3-5497 
3-5457 
3.5418 

3-5379 
3-5339 
3-5300 

0.9628 
0.9627 
0.9626 
0.9625 
0.9625 
0.9624 
0.9623 
0.9622 
0.9621 

19 

1  8 

17 
16 
15 
14 
13 

12 
II 

50 

0.2728 

0.2836 

3-5261 

0.9621 

10 

5i 
52 

53 
54 
55 
56 

57 
58 
59 

0.2731 
0.2734 
0.2737 
0.2740 
0.2742 
0.2745 
0.2748 
0.2751 
0.2754 

0.2839 
0.2842 
0.2845 
0.2849 
0.2852 
0.2855 
0.2858 
0.2861 
0.2864 

3-5222 
3-5183 
3.5H4 
3-5105 
3.5067 
3.5028 

3.4989 

3-4951 
3.4912 

0.9620 
0.9619 
0.9618 
0.9617 
0.9617 
0.9616 
0.9615 
0.9614 
0.9613 

9 

8 

7 
6 
5 
4 
3 

2 

I 

00 

0.2756 

0.2867 

3.48/4 

0.9613 

0 

Cos 

Cot 

Tan 

Sin 

' 

*165°  255°  *345°   75' 


NATURAL 


*164°  254°  *344< 


116 

*106° 


196°  *286C 


16C 


NATURAL 


17°    *107°    197°  *287< 


/ 

Sin 

Tan 

Cot 

Cos 

0 

0.2756 

0.2867 

3-4874 

0.9613 

60 

I 
2 
3 
4 

6 

7 
8 

9 

0.2759 
0.2762 
0.2765 
0.2768 
0.2770 
0.2773 
0.2776 
0.2779 
0.2782 

0.2871 
0.2874 
0.2877 
0.2880 
0.2883 
0.2886 
0.2890 
0.2893 
0.2896 

3-4836 
34798 
3.4760 

3-4722 
3-4684 
3.4646 

3.4608 
3-4570 
3-4533 

0.9612 
0.9611 
0.9610 
0.9609 
0.9609 
0.9608 
0.9607 
0.9606 
0.9605 

59 

58 
57 
56 
55 
54 

53 
52 
5i 

10 

0.2784 

0.2899 

3-4495 

0.9605 

50 

ii 

12 
13 
14 
15 

16 

17 

18 

19 

0.2787 
0.2790 
0.2793 
0.2795 
0.2798 
0.2801 
0.2804 
0.2807 
0.2809 

0.2902 
0.2905 
0.2908 
0.2912 
0.2915 
0.2918 
0.2921 
0.2924 
0.2927 

34458 
3.4420 
34383 
3-4346 
3.4308 
3.4271 

34234 
3-4I97 
3.4160 

0.9604 
0.9603 
0.9602 
0.9601 
0.9600 
0.9600 

0.9599 
0.9598 

0.9597 

49 

48 

47 
46 
45 
44 

43 
42 
41 

20 

0.2812 

0.2931 

3.4124 

0.9596 

40 

21 
22 
23 
24 

25 
26 

27 
28 
29 

0.2815 
0.2818 
0.2821 
0.2^23 
0.2826 
0.2829 
0.2832 
0.2835 
0.2837 

0.2934 
0.2937 
0.2940 
0.2943 
0.2946 
0.2949 
0.2953 
0.2956 
0.2959 

3.4087 
34050 
3.4014 

3-3977 
3-3941 
3-39°4 
3-3868 
3-3832 
3.3796 

0.9596 
0-9595 
0.9594 

o.9593 
0.9592 
0.9591 
0.9591 
0.9590 
0.9589 

39 
38 
37 
36 
35 
34 

33 
32 
31 

30 

0.2840 

0.2962 

3-3759 

0.9588 

30 

3i 

32 

33 
34 
35 
36 

37 
38 
39 

0.2843 
0.2846 
0.2849 
0.2851 
0.2854 
0.2857 
0.2860 
0.2862 
0.2865 

0.2965 
0.2968 
0.2972 
0.2975 
0.2978 
0.2981 
0.2984 
0.2987 
0.2991 

3.3723 
3.3687 
3-3652 
3-3616 
3.358o 
3-3544 
3.3509 
3-3473 
3-3438 

0.9587 
0.9587 
0.9586 

0.9585 
0.9584 

0-9583 
0.9582 
0.9582 
0.9581 

29 
28 
27 
26 
25 
24 
23 

22 
21 

40 

0.2868 

0.2994 

3.3402 

0.9580 

20 

4i 
42 

43 
44 
45 
46 

47 
48 
49 

0.2871 
0.2874 
0.2876 
0.2879 
0.2882 
0.2885 
0.2888 
0.2890 
0.2893 

0.2997 
0.3000 
0.3003 
0.3006 
0.3010 
0.3013 
0.3016 
0.3019 
0.3022 

3-3367 
3-3332 
3.3297 
3.3261 
3.3226 
3.3I91 
3.3156 
3-3122 
3-3087 

0.9579 
0.9578 

0-9577 

o.9577 
0.9576 
0-9575 
0-9574 
0-9573 
0.9572 

19 

18 

17 
16 
15 

14 
13 

12 

II 

50 

0.2896 

0.3026 

3-3052 

0.9572 

10 

51 

52 
53 
54 
55 
56 

57 
58 
59 

0.2899 
0.2901 
0.2904 
0.2907 
0.2910 
0.2913 
0.2915 
0.2918 
0.2921 

0.3029 
0.3032 
0.3035 
0.3038 
0.3041 
0.304! 
0.3048 
0.3051 
0.3054 

3-3017 
3-2983 
3.2948 

3.2914 
3.2879 
3-2845 
3.2811 
3-2777 
3-2743 

0.9571 
0.9570 
0.9569 
0.9568 
0.9567 
0.9566 
0.9566 
0.9565 
0.9564 

9 

8 

7 
6 
5 
4 
3 

i 

60 

0.2924 

0.3057 

3.2709 

0.9563 

0 

Cos 

Cot 

Tan 

Sin 

' 

' 

Sin 

Tan 

Cot 

Cos 

0 

0.2924 

0.3057 

3.2709 

0.9563 

60 

I 

2 

3 
4 

6 

7 
8 

9 

0.2926 
0.2929 
0.2932 

0.2935 
0.2938 
0.2940 
0.2943 
0.2946 
0.2949 

0.3060 
0.3064 
0.3067 
0.3070 
0.3073 
0.3076 
0.3080 
0.3083 
0.3086 

3-2675 
3-2641 
3.2607 

3-2573 
3-2539 
3.2506 

3-2472 

3-2438 
3-2405 

0.9562 
0.9561 
0.9560 
0.9560 

0.9559 
0-9558 

0-9557 
0.9556 
0-9555 

59 

58 
57 
56 

55 
54 

53 
52 
5i 

10 

0.2952 

0.3089 

3-2371 

0-9555 

50 

ii 

12 
13 
14 

15 

16 

17 

18 

*9 

0.2954 
0.2957 
0.2960 
0.2963 
0.2965 
0.2968 
0.2971 
0.2974 
0.2977 

0.3092 
0.3096 
0.3099 
0.3102 
0.3105 
0.3108 

0.3111 

0.3115 
0.3118 

3-2338 
3-2305 
3.2272 

3-2238 
3.2205 
3-2172 

3.2139 
3.2106 
3.2073 

0-9554 
0-9553 
0.9552 

o.955i 
0-9550 
0-9549 
0.9548 
0.9548 
0.9547 

49 

48 

47 
46 
45 
44 

43 
42 
4i 

20 

0.2979 

0.3121 

3-2041 

0.9546 

40 

21 
22 
23 

24 

25 
26 

27 
28 
29 

0.2982 
0.2985 
0.2988 
0.2990 
0.2993 
0.2996 
0.2999 
0.3002 
0.3004 

0.3124 
0.3127 
0.3131 
0.3134 
0.3137 
0.3140 

0.3143 
0.3147 
0.3150 

3.2008 
3.1975 
3.I943 
3.1910 

3-1878 
3-1845 

3-1813 
3.1780 
3-1748 

0-9545 
0-9544 
0-9543 
0.9542 
0.9542 
0.9541 

0.9540 
0-9539 
0.9538 

39 
38 
37 
36 
35 
34 

33 
32 
3i 

30 

0.3007 

0-3153 

3.1716 

0-9537 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 

0.3010 
0.3013 
0.3015 
0.3018 
0.3021 
0.3024 
0.3026 
0.3029 
0.3032 

0.3156 

0-3159 
0.3163 

0.3166 
0.3169 
0.3172 

0.3175 
0.3179 
0.3182 

3-1684 
3-1652 
3.1620 

3-1588 
3.1556 
3-1524 

3.1492 
3.1460 
3-1429 

0.9536 
0-9535 
0.9535 
0-9534 
0-9533 
0-9532 

o.953i 
0.9530 
0.9529 

29 
28 
27 
26 

25 
24 

23 

22 

40 

0-3035 

0.3185 

3-1397 

0.9528 

20 

4i 
42 
43 
44 
45 
46 

47 
48 
49 

0.3038 
0.3040 
0.3043 
0.3046 
0.3049 
0.3051 

0.3054 
0.3057 
0.3060 

0.3188 
0.3191 
0.319! 
0.3198 
0.3201 
0.3204 
0.3207 
0.3211 
0.3214 

3.1366 
3-1334 
3-1303 
3.1271 
3.1240 
3-1209 
3.1178 
3.1146 
3-1115 

0-9527 
0.9527 
0.9526 

0.9525 
0.9524 

0.9523 
0.9522 
0.9521 
0.9520 

19 

18 

17 

16 

15 
14 
13 

12 
II 

50 

0.3062 

0.3217 

3.1084 

0.9520 

10 

5i 

52 
53 
54 

55 
56 

57 
58 
59 

0.3065 
0.3068 
0.3071 
0.3074 
0.3076 
0.3079 
0.3082 
0.3085 
0.3087 

0.3220 
0.3223 
0.3227 
0.3230 

0.3^33 
0.3236 

0.3240 

0.3243 
0.3246 

3-1053 
3.1022 
3.0991 
3.0961 
3-0930 
3.0899 

3.0868 
3.0838 
3.0807 

0.9519 
0.9518 
0.9517 
0.9516 

0.9515 
0.9514 

0.9513 
0.9512 
0.9511 

9 

8 

7 
6 
5 
4 
3 

i 

60 

0.3090 

0.3249 

3.0777 

0.9511 

0 

Cos 

Cot 

Tan 

Sin 

' 

*163°  253°  *343 


NATURAL 


72°      *162°  252°  *342C 


*108°  198°  *288° 


NATURAL 


19C 


117 

199°  *289° 


' 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 
4 
5 
6 

7 
8 

9 
10 
ii 

12 
13 
14 
15 

1  6 

17 
18 

T9 
20 

21 
22 
23 
24 

25 
26 

27 
28 
29 

30 

3i 
32 
33 
34 

35 
36 

37 
38 
39 
40 

4i 
42 
43 
44 

45 
46 

47 

48 

49 
50 

5i 

52 
53 
54 
55 
56 

57 
58 
59 
60 

0.3090 

0.3249 

3-0777 

0.9511 

60 

59 

58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 
47 
46 
45 
44 

43 
42 

4i 
40 

39 

38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 
28 
27 
26 

25 
24 

23 

22 
21 
20 
19 

18 
17 
16 
15 
14 
13 

12 
II 

10 

9 
8 

7 
6 
5 
4 
3 

2 

I 

0 

0.3093 
0.3090 
0.3098 
0.3101 
0.3104 
0.3107 
0.3110 
0.3112 
o.3"5 

0.3252 
0.3256 
0.3259 
0.3262 
0.3265 
0.3269 
0.3272 
0.3275 
0.3278 

3.0746 
3-0716 
3.0686 
3-0655 
3.0625 
3.0595 
3-0565 
3-0535 
3-0505 

0.9510 
0.9509 
0.9508 
0.9507 
0.9506 
0.9505 
0.9504 

0.9503 
0.9502 

0.3118 

0.3281 

3-0475 

0.9502 

0.3121 
0.3123 
0.3126 
0.3129 
0.3132 
0.3134 

0.3137 
0.3140 

9-3143 

0.3285 
0.3288 
0.3291 
0.3294 
0.3298 
0.3301 
0.3304 

0.3307 
0.3310 

3-0445 
3-0415 
3-0385 
3.0356 
3.0326 
3.0296 
3.0267 
3-0237 
3.0208 

0.9501 
0.9500 
0.9499 
0.9498 

0-9497 
0.9496 

0.9495 
o.9494 
0.9493 

o.3M5 

0.3314 

3.0178 

0.9492 

0.3148 
0.3151 
0.3154 
03156 

03159 
0.3162 

0.3165 
0.3168 
0.3170 

0.3317 
0.3320 

0.3323 

0.3327 
0.3330 
0-3333 
0.3336 
0.3339 
0.3343 

3.0149 
3.0120 
3.0090 
3.0061 
3.0032 
3.0003 
2.9974 
2.9945 
2.9916 

0.9492 
0.9491 
0.9490 
0.9489 
0.9488 
0.9487 
0.9486 
0.9485 
0.9484 

0.3173 

o  334& 

2.9887 

0.9483 

0.3176 

0.3179 
0.3181 

0.3184 
0.3187 
0.3190 
0.3192 

0.3195 
0.3198 

0.3349 
0.3352 
0.3356 

0.3359 
0.3362 

0.3365 
0.3369 
0.3372 
0.3375 

2.9858 
2.9829 
2.9800 
2.9772 

2.9743 
2.9714 

2.9686 
2.9657 
2.9629 

0.9482 
0.9481 
0.9480 
0.9480 

0.9479 
0.9478 

o.9477 
0.9476 

0-9475 

0.3201 

0.3378 

2.9600 

0.9474 

0.3203 
0.3206 
0.3209 
0.3212 
0.3214 
0.3217 

0.3220 
0.3223 
0.3225 

0.3382 
0.3385 

0.3388 

0.3391 
0.3395 
0.3398 
0.3401 
0.3404 

0.3408 

2.9572 
2-9544 
2.9515 
2.9487 
2-9459 
2-9431 
2.9403 
2-9375 
2-9347 

0.9473 
0.9472 
0.9471 

0.9470 
0.9469 
0.9468 
0.9467 
0.9466 
0.9466 

0.3228 

0.3411 

2.9319 

0.9465 

0.3231 

0.3234 
0.3236 

0.3239 
0.3242 

0.3245 

0.3247 
0.3250 

0.3253 

-0.3414 

0.3417 
0.3421 

0.3424 
0.3427 
0.3430 

0.3434 
0.3437 
0.3440 

2.9291 
2.9263 
2.9235 
2.9208 
2.9180 
2.9152 
2.9125 
2.9097 
2.9070 

0.9464 
0.9463 
0.9462 
0.9461 
0.9460 
0.9459 
0.9458 

o.9457 
0.9456 

0.3256 

0.3443 

2.9042 

0.9455 

Cos 

Cot 

Tan 

Sin 

' 

161C  251°  *341°       71* 


NATURAL 


' 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 

4 
5 
6 

8 
9 
10 
ii 

12 
13 
14 

15 

16 

17 

18 

J9 
20 

21 
22 
23 
24 

25 
26 

27 
28 
29 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 
42 
43 
44 
45 
46 

47 
48 
49 
50 

5i 

52 
53 
54 

55 
56 

57 
58 
59 
60 

0.3256 

0.3443 

2.9042 

0.9455 

60 
59 

57 
56 
55 
54 

53 
52 
51 
50 

49 

48 

47 
46 
45 
44 

43 
42 

4i 
40 

39 
38 
37 
36^ 
35 
34 

.33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 

22 
21 

20 

19 

18 

17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.3258 
0.3261 
0.3264 
0.3267 
0.3269 
0.3272 

0.327! 
0.3278 
0.3280 

o.3447 
0.3450 
o.3453 
0.3456 
0.3460 
0.3463 
0.3466 
0.3469 
0-3473 

2.9015 
2.8987 
2.8960 

2.8933 
2.8905 
2.8878 
2.8851 
2.8824 
2.8797 

0.9454 
0.9453 
0.9452 

0.9451 
0.9450 

0.9449 
0.9449 
0.9448 
0.9447 

0.3283 

0.3476 

2.8770 

0.9446 

0.3286 
0.3289 
0.3291 
0.3294 

0.3297 
0.3300 

0.3302 

0.3305 
0.3308 

0.3479 
0.3482 
0.3486 
0.3489 
0.3492 
o.3495 
0.3499 
0.3502 
0.3505 

2.8743 
2.8716 
2.8689 
2.8662 
2.8636 
2.8609 
2.8582 
2.8556 
2.8529 

o.9445 
0.9444 
0.9443 

0.9442 
0.9441 
0.9440 

0.9439 
0.9438 
0.9437 

0.3311 

0.3508 

2.8502 

0.9436 

0.3313 
0.3316 

0.3319 

0.3322 
0.3324 
0.3327 
0.3330 
0-3333 
0-3335 

0.3512 
0.3515 
0.3518 

0.3522 
0.3525 
0.3528 

0.3531 
0-3535 
0.3538 

2.8476 
2.8449 
2.8423 
2.8397 
2.8370 
2.8344 
2.8318 
2.8291 
2.8265 

0.9435 
0.9434 

0.9433 
0.9432 
0.9431 
0.9430 

0.9429 
0.9428 
0.9427 

0.3338 

0.3541 

2.8239 

0.9426 

0.3341 
o.3344 
0.3346 

o.3349 
0.3352 
0-3355 

0.3357 
0.3360 

0.3363 

0-3544 
0.3548 
o.355i 
0.3554 
0.3558 
0.3561 

0.3564 
0.3567 
0.3571 

2.8213 
2.8187 
2.8161 
2.8135 
2.8109 
2.8083 
2.8057 
2.8032 
2.8006 

0.9425 

0.9424 
0.9423 

0.9423 
0.9422 
0.9421 
0.9420 
0.9419 
0.9418 

0.3365 

0-3574 

2.7980 

0.9417 

0.3368 
0.3371 
o.3374 
0.3376 

0-3379 
0.3382 

0.3385 
0.3387 
0.3390 

0-3577 
0.3581 
0.3584 
0.3587 
0.359° 
0.3594 

0.3597 
0.3600 
0.3604 

2-7955 
2.7929 

2.7903 
2.7878 
2.7852 
2.7827 
2.7801 
2,7776 
2-7751 

0.9416 
0.9415 
0.9414 
0.9413 
0.9412 
0,9411 
0.9410 
0.9409 
0.9408 

Q.3393 
0.3396 
0.3398 
0.3401 

0.3404 

0.3407 
0.3409 

0.3412 

0.3415 
0.3417 

0.3607 

2.7725 

0.9407 

0.3610 
0.3613 
0.3617 
0.3620 
0.3623 
0.3627 
0.3630 

0.3633 
0.3636 

2.7700 

2.76/5 
2.7650 

2.7625 
2.7600 
2.7575 
2.7550 
2.7525 
2.7500 

0.9406 
0.9405 
0.9404 
0.9403 
0.9402 
0.9401 
0.9400 

0.9399 
0.9398 

0.3420  |  0.3640 

2.7475 

0.9397 

Cos    Cot   Tan 

Sin 

' 

70C 


*160°  250°  *340C 


118 

*110°  200°  *290C 


20C 


NATURAL 


21°   *111°  201°  *291< 


' 

Sin 

Tan 

Cot 

Cos 

0 

0.3420 

0.3640 

2-7475 

0.9397 

GO 

I 

2 

3 
4 

5 
6 

7 
8 

9 

0.3423 
0.3426 
0.3428 

0.3431 
0-3434 
0-3437 

0-3439 
0.3442 
o.3445 

0.3643 
0.3646 
0.3650 

0.3653 
0.3656 

0.3659 
0.3663 
0.3666 
0.3669 

2.7450 
2.7425 
2.7400 

2.7376 
2.7351 
2.7326 

2.7302 

2.7277 
2.7253 

0.9396 
0.939! 
0-9394 

0.9393 
0.9392 
0.9391 

0.9390 
0.9389 
0.9388 

59 

58 
57 
56 
55 
54 
53 
52 
51 

10 

0.3448 

0.3673 

2.7228 

0.9387 

50 

ii 

12 
13 
14 
15 

16 

17 

18 

19 

0.3450 
0-3453 
0.3456 

0.3458 
0.3461 
0.3464 
0.3467 
0.3469 
0.3472 

0.3676 

0.3679 
0.3683 

0.3686 
0.3689 
0.3693 
0.3696 
0.3699 
0.3702 

2.7204 
2.7179 
2.715! 
2.7130 
2.7106 
2.7082 
2.7058 
2.7034 
2.7009 

0.9386 

0.9385 
0.9384 

0.9383 
0.9382 
0.9381 
0.9380 

0.9379 
0.93/8 

49 
48 
47 
46 
45 
44 

43 
42 

4i 

20 

0-3475 

0.3706 

2.6985 

0-9377 

40 

21 

22 
23 
24 
25 
26 

27 
28 

29 

0.3478 
0.3480 
0.3483 
0.3486 
0.3488 
0.3491 

0-3494 
0-3497 
0-3499 

0.3709 
0.3712 
0.3716 
0.3719 
0.3722 
0.3726 
0.3729 
0.3732 
0.3736 

2.6961 

2.6937 
2.6913 

2.6889 
2.6865 
2.6841 
2.6818 
2.6794 
2.6770 

0.9376 
0-9375 
o.9374 

0-9373 
0.9372 
0.9371 
0.9370 
0.9369 
0.9368 

39 
38 
37 
36 
35 
34 

33 
32 
3i 

30 

0.3502 

0.3739 

2.6746 

0.9367 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 

0.3505 
0.3508 
0.3510 

0.3513 
0.3516 
0.3518 

0.3521 
0.3524 
0.3527 

0.3742 
0-3745 
o.3749 
0.3752 
0-3755 
o.3759 
0.3762 
0.3765 
0.3769 

2.6723 
2.6699 
2.6675 
2.6652 
2.6628 
2.6605 
2.6581 
2.6558 
2.6534 

0.9366 
0.9365 
0.9364 

0.9363 
0.9362 
0.9361 
0.9360 

0.9359 
0.9358 

29 

28 
27 
26 
25 
24 
23 

22 
21 

40 

0.3529 

0.3772 

2.6511 

0.9356 

20 

4i 
42 
43 
44 

45 
46 

47 

48 

49 

0.3532 
0.3535 
0-3537 
0.3540 
0-3543 
0.3546 

0.3548 
o.355i 
0-3554 

0-3775 
0.3779 
0.3782 

0.3785 
0.3789 
0.3792 

0.3795 
0.3/99 
0.3802 

2.6488 
2.6464 
2.6441 
2.6418 
2.6395 
2.6371 

2.6348 
2.6325 

2.0302 

o.9355 
o.9354 
o.9353 
0.9352 
0-9351 
0.9350 
0.9349 
0.9348 
0.9347 

19 

18 

17 
16 
15 
14 
13 

12 
II 

50 

0-3557 

0.3805 

2.6279 

0.9346 

10 

5i 
52 
53 
54 
55 
56 

57 
58 
59 

0-3559 
0.3562 

0.356! 
0.3567 
0.3570 
0-3573 
0.3576 
0.3578 
0.3581 

0.3809 
0.3812 
0.3815 
0.3819 
0.3822 
0.3825 
0.3829 
0.3832 
0.3835 

2.6256 
2.6233 
2.6210 
2.6187 
2.6165 
2.6142 
2.6119 
2.6096 
2.6074 

0-9345 
0.9344 
o.9343 
0.9342 
0.9341 
0.9340 

0.9339 
0.9338 
0-9337 

9 

8 

7 
6 
5 
4 
3 

2 
I 

GO 

0.35*4 

0.3839 

2.6051 

0.9336 

0 

Cos 

Cot 

Tan 

Sin 

' 

Sin 

Tan 

Cot 

Cos 

0 

0.3584 

0.3839 

2.6051 

0.9336 

GO 

I 

2 

3 
4 
5 
6 

8 
9 

0.3586 
0.3589 
0.3592 
0.3595 
0.3597 
0.3600 

0.3603 
0.3605 
0.3608 

0.3842 
0.3845 
0.3849 
0.3852 
0.3855 
0.3859 
0.3862 
0.3865 
0.3869 

2.6028 
2.6006 
2.5983 
2.5961 
2.5938 
2.5916 

2.5893 
2.5871 
2.5848 

0.933! 
0.9334 
0-9333 
0.9332 

0.9331 
0.9330 

0.9328 

0.9327 
0.9326 

59 
58 
57 
56 
55 
54 

53 

52 
5i 

10 

0.3611 

0.3872 

2.5826 

0.9325 

50 

ii 

12 
13 
14 
15 

16 

17 

18 

19 

0.3614 
0.3616 
0.3619 
0.3622 
0.3624 
0.3627 
0.3630 
0.3633 
0.3635 

0.3875 
0.3879 
0.3882 
0.3885 
0.3889 
0.3892 

0.3895 
0.3899 
0.3902 

2.5804 
2.5782 
2-5759 
2-5737 
2.5715 
2.5693 
2.5671 
2.5649 
2.5627 

0.9324 
0.9323 
0.9322 
0.9321 
0.9320 
0.9319 
0.9318 
0.9317 
0.9316 

49 

48 

47 
46 
45 
44 

43 
42 
4i 

20 

0.3638 

0.3906 

2.5605 

0.9315 

40 

21 

23 
24 

25 
26 

27 
28 

29 

0.3641 
0.3643 
0.3646 

0.3649 
0.3651 
0.3654 

0.3657 
0.3660 
0.3662 

0.3909 
0.3912 
0.3916 
0.3919 
0.3922 
0.3926 

0.3929 
0.3932 
0.3936 

2.5583 
2.5561 
2-5539 
2.5517 
2-5495 
2-5473 
2.5452 
2.5430 
2.5408 

0.9314 

0.9313 
0.9312 

0.9311 
0.9309 
0.9308 

0.9307 
0.9306 
0.9305 

39 
38 
37 
36 
35 
34 

33 
32 
3i 

30 

0.3665 

0.3939 

2.5386 

0.9304 

30 

31 
32 
33 
34 
35 
36 

37 
38 
39 

0.3668 
0.3670 
0.3673 
0.3676 
0.3679 
0.3681 
0.3684 
0.3687 
0.3689 

0.3942 
0.3946 
0.3949 

0-3953 
0.3956 
0-3959 
0.3963 
0.3966 
0.3969 

2.5365 
2-5343 
2.5322 

2.5300 
2.5279 
2-5257 
2.5236 
2.5214 
2.5193 

0.9303 
0.9302 
0.9301 
0.9300 
0.9299 
0.9298 

0.9297 
0.9296 

0.9295 

29 

28 
27 
26 

25 
24 

23 

21 

40 

0.3692 

0-3973 

2.5172 

0.9293 

20 

4i 
42 
43 
44 
45 
46 

47 

48 

49 

0.369! 
0.3697 
0.3700 

0.3703 
0.3706 
0.3708 
0.3711 

0.3714 
0.3716 

0.3976 
0-3979 
o.3983 
0.3986 
0.3990 
o.3993 
0.3996 
0.4000 
0.4003 

2.5150 
2.5129 
2.5108 
2.5086 
2.5065 
2.5044 
2.5023 
2.5002 
2.4981 

0.9292 
0.9291 
0.9290 
0.9289 
0.9288 
0.9287 
0.9286 
0.9285 
0.9284 

!9 

18 
17 
1  6 
15 
14 
13 

12 
II 

50 

0.3719 

0.4006 

2.4960 

0.9283 

10 

5i 
52 
53 
54 
55 
56 

57 

58 
59 

0.3722 
0.3724 
0.3727 
0.3730 
0-3733 
0-3735 
0.3738 
0.3741 
0.3743 

0.4010 
0.4013 
0.4017 
0.4020 
0.4023 
0.4027 
0.4030 
0.4033 
0.4037 

2-4939 
2.4918 
2.4897 
2.4876 
2.485! 
2.4834 
2.4813 
2.4792 
2.4772 

0.9282 
0.9281 
0.9279 
0.9278 
0.9277 
0.9276 
0.9275 
0.9274 
0.9273 

9 

8 

7 
6 
5 
4 
3 

I 

60 

0.3746 

0.4040 

2.4751 

0.9272 

0 

Cos 

Cot 

Tan 

Sin 

' 

*159°  249°  *339° 


NATURAL 


08' 


*158°  248°  *338< 


*112°  202°  *292°        22° 


NATURAL 


23° 


119 
*113°  203°  *293° 


' 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 
4 
5 
6 

7 
8 

9 
10 

0.3746 

0.4040 

2.4751 

0.9272 

60 

59 
58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 

28 

27 
26 
25 
24 
23 

22 
21 

20 

19 

18 

17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0-3749 
o.375i 
0.3754 

0-3757 
0.3760 
0.3762 

0.3765 
0.3768 
0.3770 

0.4044 
0.4047 
0.4050 
0.4054 
0.4057 
0.4061 
0.4064 
0.4067 
0.4071 

2.4730 
2.4709 
2.4689 
2.4668 
2.4648 
2.4627 
2.4606 
2.4586 
2.4566 

0.9271 
0.9270 
0.9269 
0.9267 
0.9266 
0.9265 
0.9264 
0.9263 
0.9262 

0-3773 

0.4074 

2-4545 

0.9261 

ii 

12 
13 
14 
15 

16 

17 

18 

19 
20 

0.3776 
0.3778 
0.3781 
0.3784 
0.3786 
0.3789 

0.3792 
0-3795 
0.3797 

0.4078 
0.4081 
0.4084 
0.4088 
0.4091 
0.4095 
0.4098 
0.4101 
0.4105 

2.4525 
2.4504 
2.4484 
2.4464 
2.4443 
2.4423 
2.4403 

2.4383 
2.4362 

0.9260 
0.9259 
0.9258 

0.9257 
0.9255 
0.9254 

0.9253 
0.9252 
0.9251 

0.3800 

0.4108 

2.4342 

0.9250 

21 

22 
23 
24 

25 
26 

27 
28 
29 

0.3803 
0.3805 
0.3808 
0.3811 
0.3813 
0.3816 
0.3819 
0.3821 
0.3824 

0.4111 
0.4115 
0.4118 

0.4122 
0.4125 
0.4129 

0.4132 
0.4135 
0.4139 

2.4322 
2.4302 
2.4282 
2.4262 
2.4242 
2.4222 
2.4202 
2.4182 
2.4162 

0.9249 
0.9248 
0.9247 
0.9245 
0.9244 
0.9243 
0.9242 
0.9241 
0.9240 

30 

0.3827 

0.4142 

2.4142 

0.9239 

31 

32 

33 
34 

35 
36 

37 
38 
39 

0.3830 
0.3832 
0.3835 
0.3838 
0.3840 
0.3843 
0.3846 
0.3848 
0.3851  • 

0.4146 
0.4149 
0.4152 
0.4156 
0.4159 
0.4163 
0.4166 
0.4169 
0.4173 

2.4122 
2.4102 
2.4083 
2.4063 
2.4043 
2.4023 
2.4004 

2.3984 
2.3964 

0.9238 
0.9237 
0.9235 
0.9234 
0.9233 
0.9232 
0.9231 
0.9230 
0.9229 

40 

0.3854 

0.4176 

2.3945 

0.9228 

4i 
42 

43 
44 
45 
46 

47 
48 
49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 
60 

0.3856 

0.3859 
0.3862 

0.3864 
0.3867 
0.3870 

0.3872 
0.3875 
0.3878 

0.4180 
0.4183 
0.4187 
0.4190 
0.4193 
0.4197 
0.4200 
0.4204 
0.4207 

2.3925 
2.3906 
2.3886 
2.3867 

2.3847 
2.3828 

2.3808 
2.3789 
2.3770 

0.9227 
0.9225 
0.9224 
0.9223 
0.9222 
0.9221 
0.9220 
0.9219 
0.9218 

0.3881 

0.4210 

2-3750 

0.9216 

0.3883 
0.3886 
0.3889 
0.3891 
0.3894 
0.3897 

0.3899 
0.3902 
0.3905 

0.4214 
0.4217 
0.4221 
0.4224 
0.4228 
0.4231 
0.4234 
0.4238 
0.4241 

2-3731 
2.3712 
2.3693 

2.3673 
2.3654 
2.363! 
2.3616 
2.3597 
2.3578 

0.9215 
0.9214 
0.9213 
0.9212 
0.9211 
0.9210 
0.9208 
0.9207 
0.9206 

0.3907 

0.4245 

2-3559 

0.9205 

Cos  |  Cot 

Tan 

Sin 

' 

*157°  247°  *337°   67' 


NATURAL 


' 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 

6 

8 
9 
10 
ii 

12 
13 
14 
15 

16 

17 

18 

J9 
20 

0.3907 

0.4245 

2.3559 

0.9205 

60 

59 

58 
57 
56 
55 
54 

53 
52 
51 
50 

49 

48 

47 
46 
45 
44 

43 
42 

41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 

22 
21 
20 
!9 

18 
17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0.3910 
0.3913 
0.39*5 
0.3918 
0.3921 
0.3923 
0.3926 
0.3929 
o.393i 

0.4248 
0.4252 
0.4255 
0.4258 
0.4262 
0.4265 
0.4269 
0.4272 
0.4276 

2.3539 
2.3520 
2.3501 

2.3483 
2.3464 
2-3445 
2.3426 
2.3407 
2.3388 

0.9204 
0.9203 
0.9202 
0.9200 
0.9199 
0.9198 
0.9197 
0.9196 
0.9195 

0-3934 

0.4279 

2.3369 

0.9194 

0.3937 
0-3939 
0.3942 

0.3945 
0-3947 
o.395o 

0-3953 
0-3955 
0.3958 

0.4283 
0.4286 
0.4289 

0.4293 
0.4296 
0.4300 
0.4303 
0.4307 
0.4310 

2-3351 
2.3332 
2.3313 

2.3294 
2.3276 

2.3257 
2.3238 
2.3220 
2.3201 

0.9192 
0.9191 
0.9190 
0.9189 
0.9188 
0.9187 
0.9186 
0.9184 
0.9183 

0.3961 

0.4314 

2.3183 

0.9182 

21 
22 
23 
24 

25 
26 

27 

28 

29 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 
42 
43 
44 
45 
46 

47 
48 

49 
50 

5i 

52 
53 
54 
55 
56 

57 
58 
59 
60 

0.3963 
0.3966 
0.3969 
0.3971 
0-3974 
0.3977 

0-3979 
0.3982 

o.3985 

0.4317 
0.4320 
0.4324 
0.4327 
0.4331 
0-4334 
0.4338 
0.4341 
0-4345 

2.3164 
2.3146 
2.3127 
2.3109 
2.3090 
2.3072 

2.3053 
2.3035 
2.3017 

0.9181 
0.9180 
0.9179 
0.9178 
0.9176 
0.9175 
0.9174 

0.9173 
0.9172 

0.3987 

0.4348 

2.2998 

0.9171 

0.399° 
0.3993 
0.3995 

0.3998 
0.4001 
0.4003 
0.4006 
0.4009 
0.4011 

0.4352 
0.4355 
0.4359 
0.4362 

0.4365 
0.4369 

0.4372 
0.4376 

0.4379 

2.2980 
2.2962 
2.2944 
2.2925 
2.2907 
2.2889 
2.2871 
2.2853 
2.2835 

0.9169 
0.9168 
0.9167 
0.9166 
0.9165 
0.9164 
0.9162 
0.9161 
0.9160 

0.4014 

0.4383 

2.2817 

0.9159 

0.4017 
0.4019 
0.4022 
0.4025 
0.4027 
0.4030 

0.4033 
0.4035 
0.4038 

0.4386 
0.439° 
0.4393 

0.4397 
0.4400 
0.4404 
0.4407 
0.4411 
0.4414 

2.2799 
2.2781 
2.2763 
2.2745 
2.2727 
2.2709 
2.2691 
2.2673 
2.2655 

0.9158 
0.9157 
0.9155 
0.9154 

0.9153 
0.9152 

0.9151 
0.9150 
0.9148 

0.4041 

0.4417 

2.2637 

0.9147 

0.4043 
0.4046 
0.4049 
0.4051 

0.4054 
0.4057 

0.4059 
0.4062 
0.4065 

0.4421 
0.4424 
0.4428 

0.4431 
0.4435 
0.4438 

0.4442 
0.4445 
0.4449 

2.2620 
2.2602 
2.2584 
2.2566 
2.2549 
2.2531 
2.2513 
2.2496 
2.2478 

0.9146 
0.9145 
0.9144 

0.9143 
0.9141 
0.9140 
0.9139 
0.9138 
0.9137 

0.4067 

0.4452 

2.2460 

0.9135 

0 

Cos 

Cot 

Tan 

Sin 

' 

66° 


246°  *336° 


120 

*114°  204°  *294° 


24C 


NATURAL 


25< 


*115°  205°  *295° 


' 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 

4 
5 
6 

7 
8 

9 
10 
ii 

12 
13 
14 
15 

16 

17 

18 

19 

20 

21 
22 
23 
24 

25 
26 

27 
28 

29 

0.4067 

0.4452 

2.2460 

0.9135 

00 

59 
58 
57 
56 

55 
54 

53 
52 
5i 
50 

49 

48 

47 
46 
45 
44 

43 
42 

4i 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 

28 

27 
26 

25 
24 

23 

22 
21 

20 

19 

18 

17 
16 
15 
14 
13 

12 
II 
10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.4070 
0.4073 
0.4075 
0.4078 
0.4081 
0.4083 
0.4086 
0.4089 
0.4091 

0.4456 
0.4459 
0.4463 
0.4466 
0.4470 
0.4473 
0.4477 
0.4480 
0.4484 

2.2443 
2.2425 
2.2408 
2.2390 
2.2373 
2.2355 
2.2338 
2.2320 
2.2303 

0.9134 

0.9133 
0.9132 

0.9131 
0.9130 
0.9128 
0.9127 
0.9126 
0.9125 

0.4094 

0.4487 

2.2286 

0.9124 

0.4097 
0.4099 
0.4102 
0.4105 
0.4107 
0.4110 
0.4112 
0.4115 
0.4118 

0.4491 
0.4494 
0.4498 
0.4501 
0.4505 
0.4508 

0.4512 

0.4515 
0.4519 

2.2268 
2.2251 
2.2234 
2.2216 
2.2199 
2.2182 
2.2165 
2.2148 
2.2130 

0.9122 
0.9121 
0.9120 
0.9119 
0.9118 
0.9116 
0.9115 
0.9114 
0.9113 

0.4120 

0.4522 

2.2113 

0.9112 

0.4123 
0.4126 
0.4128 

0.4131 
0.4134 
0.4136 
0.4139 
0.4142 
0.4144 

0.4526 
0.4529 
0.4533 
0.4536 
0.4540 
0.4543 

0.4547 
0.4550 
0-4554 

2.2096 
2.2079 
2.2062 
2.2045 
2.2028 

2.2OII 
2.1994 
2.1977 
2.1960 

0.9110 
0.9109 
0.9108 
0.9107 
0.9106 
0.9104 
0.9103 
0.9102 
0.9101 

3U 

0.4147 

0-4557 

2.1943 

0.9100 

31 

32 

33 
34 
35 
36 

37 
38 
39 

0.4150 
0.4152 
0.4155 
0.4158 
0.4160 
0.4163 
0.4165 
0.4168 
0.4171 

0.4561 
0.4564 
0.4568 

0.4571 
0-4575 
0.4578 

0.4582 
0.4585 
0.4589 

2.1926 
2.1909 
2.1892 
2.1876 
2.1859 
2.1842 
2.1825 
2.1808 
2.1792 

0.9098 
0.9097 
0.9096 
0.9095 
0.9094 
0.9092 
0.9091 
0.9090 
0.9089 

40 

0.4173 

0.4592 

2.1775 

0.9088 

4i 
42 
43 
44 
45 
46 

47 
48 

49 

0.4176 
0.4179 
0.4181 
0.4184 
0.4187 
0.4189 
0.4192 
0.4195 
0.4197 

0.4596 
0-4599 
0.4603 

0.4607 
0.4610 
0.4614 
0.4617 
0.4621 
0.4624 

2.1758 
2.1742 
2.1725 
2.1708 
2.1692 
2.1675 
2.1659 
2.1642 
2.1625 

0.9086 
0.9085 
0.9084 
0.9083 
0.9081 
0.9080 
0.9079 
0.9078 
0.9077 

50 

0.4200 

0.4628 

2.1609 

0.9075 

5i 
52 

53 
54 
55 
56 

57 
58 
59 

0.4202 
0.4205 
0.4208 
0.4210 
0.4213 
0.4216 

0.4218 
0.4221 
0.4224 

0.4631 

0.4635 
0.4638 

0.4642 
0.4645 
0.4649 
0.4652 
0.4656 
0.4660 

2.1592 
2.1576 
2.1560 

2.1543 
2.1527 
2.I5IO 

2.1494 
2.1478 
2.1461 

0.9074 
0.9073 
0.9072 
0.9070 
o.goOc) 
0.9068 

0.9067 
o.'j  4  6 
0.9064 

60 

0.4226 

0.4663 

2.1445 

0.9063 

Cos 

Cot 

Tan 

Sin 

' 

•155°  245°  *335C 


NATURAL 


' 

Sin 

Tan 

Cot 

Cos 

0 

0.4226 

0.4663 

2.1445 

0.9063 

60 

I 

2 

3 
4 

6 

7 
8 

9 

0.4229 
0.4231 
0.4234 

0.4237 
0.4239 
0.4242 

0.4245 
0.4247 
0.4250 

0.4667 
0.4670 
0.4674 
0.4677 
0.4681 
0.4684 
0.4688 
0.4691 
0.4695 

2.1429 
2.1413 
2.1396 
2.1380 
2.1364 
2.1348 

2.1332 

2.1315 
2.1299 

0.9062 
0.9061 
0.9059 
0.9058 
0.9057 
0.9056 
0.9054 

0.9053 
0.9052 

59 

53 
57 
56 
55 
54 

53 
52 
51 

10 

0.4253 

0.4699 

2.1283 

0.9051 

50 

ii 

12 

13 

14 

15 

16 

17 

18 

!9 

04255 
0.4258 
0.4260 
0.4263 
0.4266 
0.4268 
0.4271 
0.4274 
0.4276 

0.4702 
0.4706 
0.4709 

0.4713 
0.4716 
0.4720 
0.4723 
0.4727 
0.4731 

2.1267 
2.1251 
2.1235 
2.1219 
2.1203 
2.1187 
2.1171 
2.1155 
2.1139 

0.9050 
0.9048 
0.9047 
0.9046 
0.9045 
0.9043 
0.9042 
0.9041 
0.9040 

49 
48 

47 
46 
45 
44 

43 
42 
4i 

20 

0.4279 

0.4734 

2.1123 

0.9038 

40 

21 
22 
23 
24 

25 
26 

27 

28 
29 

0.4281 
0.4284 
0.4287 
0.4289 
0.4292 
0.4295 
0.4297 
0.4300 
0.4302 

0.4738 
0.4741 
0-4745 
04748 
6.4752 
0-4755 

0-4759 
0.4763 
0.4766 

2.1107 
2.1092 
2.1076 
2.1060 
2.1044 
2.1028 
2.1013 
2.0997 
2.0981 

0.9037 
0.9036 
0.903! 
0.9033 
0.9032 
0.9031 
0.9030 
0.9028 
0.9027 

39 
38 
37 
36 
35 
34 

33 
32 
3i 

90 

0.4305 

0.4770 

2.0965 

0.9026 

30 

31 
32 
33 
34 
35 
36 

37 
38 

OQ 

0.4308 
0.4310 
0.4313 
0.4316 
0.4318 
0.4321 

0.4323 
0.4326 

0.4329 

0.4773 
0.4777 
0.4780 
0.4784 
0.4788 
0.4791 

0-4795 
0.4798 
0.4802 

2.0950 
2.0934 
2.0918 
2.0903 
2.0887 
2.0872 
2.0856 
2.0840 
2.0825 

0.9025 
0.9023 
0.9022 
0.9021 
0.9020 
0.9018 
0.9017 
0.9016 
.0.9015 

29 

28 

27 
26 
25 
24 

23 

22 
21 

40 

o.433i 

0.4806 

2.0809 

0.9013 

20 

41 

42 

43 

44 
45 
46 

47 

48 

0.4334 
0-4337 
0.4339 
0.4342 
0.4344 
0.4347 
0.4350 
0.4352 
0-4355 

0.4809 
0.4813 
0.4816 
0.4820 
0.4823 
0.4827 
0.4831 
0.4834 
0.4838 

2.0794 
2.0778 
2.0763 
2.0748 
2.0732 
2.0717 
2.0701 
2.0686 
2.0671 

0.9012 
0.9011 
0.9010 
0.9008 
0.9007 
0.9006 
0.9004 
0.9003 
0.9002 

19 

18 

17 
16 
15 
U 
13 

12 
II 

4y 
50 

0.4358 

0.4841 

2.0655 

0.9001 

10 

5i 
52- 
53 
54 

55 
56 

57 
58 

0.4360 
0.4363 
0.4365 
0.4368 
0.4371 
0.4373 
0.4376 
0.4378 
^0.4381 

0.4845 
0.4849 
0.4852 
0.4856 
0.4859 
0.4863 
0.4867 
0.4870 
0.4874 

2.0640 
2.0625 
2.0609 
2.0594 

2.0579 
2.0564 

2.0549 
2.0533 
2.0518 

0.8999 
0.8998 
0.8997 
0.8996 
0.8994 
0.8993 
0.8992 
0.8990 
0.8989 

9 

8 

7 
6 
5 
4 
3 

2 
I 

59 

0.4384 

0.4877 

2.0503 

0.8988 

0 

Cos 

Cot 

Tan 

Sin 

' 

*154°  244°  *334° 


*116°  206°  *296°   26' 


NATURAL 


27< 


121 
*117°  207°  *297° 


' 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 

5 
6 

7 
8 

9 
10 
ii 

12 
13 
14 

15 

16 

17 
18 

19 
20 

21 

22 
23 
24 

25 
26 

27 

28 

29 

0.4384 

0.4877 

2.0503 

0.8988 

60 

59 

58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 

48 
47 
46 
45 
44 

43 
42 

4i 
40 

39 
38 
37 
36 
35 
34 

33 
32 
31 
30 

29 
28 
27 
26 

25 
24 

23 

22 
2t 

20 

J9 

18 

17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 

3 

2 
I 

0 

0.4386 

0.4389 
0.4392 
0.4394 
0.4397 
0-4399 
0.4402 
0.4405 
0.4407 

0.4881 
0.4885 
0.4888 
0.4892 
0.4895 
0.4899 

0.4903 
0.4906 
0.4910 

2.0488 
2.0473 
2.0458 
2.0443 
2.0428 
2.0413 
2.0398 
2.0383 
2.0368 

0.8987 
0.8985 
0.8984 
0.8983 
0.8982 
0.8980 
0.8979 
0.8978 
0.8976 

0.4410 

0.4913 

2.0353 

0.8975 

0.4412 
0.4415 
0.4418 
0.4420 

0.4423 
0.4425 

0.4428 
0.4431 
0.4433 

0.4917 
0.4921 
0.4924 
0.4928 
0.4931 
0-4935 
0.4939 
0.4942 
0.4946 

2.0338 
2.0323 
2.0308 
2.0293 
2.0278 
2.0263 
2.0248 
2.0233 
2.0219 

0.8974 
0.8973 
0.8971 
0.8970 
0.8969 
0.8967 
0.8966 
0.8965 
0.8964 

0.4436 

0.4950 

2.0204 

0.8962 

0.4439 
0.4441 

0-4444 
0.4446 

0-4449 
0.4452 

0.4454 
0-4457 
0.4459 

0.4953 
0-4957 
0.4960 

0.4964 
0.4968 
0.4971 

0-4975 
0.4979 
0.4982 

2.0189 
2.0174 
2.0160 
2.0145 
2.0130 
2.0115 

2.0IOI 
2.0086 
2.OO72 

0.8961 
0.8960 
0.8958 
0.8957 
0.8956 
0.8955 
0.8953 
0.8952 
0.8951 

30 

0.4462 

0.4986 

2.OO57 

0.8949 

3i 
32 
33 
34 

35 
36 

37 
38 
39 

0.4465 
0.4467 
0.4470 

0.4472 
0-4475 
0.4478 

0.4480 
0.4483 
0.4485 

0.4989 

o.4993 
0.4997 

0.5000 
0.5004 
0.5008 
0.5011 
0.5015 
0.5019 

2.OO42 
2.OO28 
2.0013 
1.9999 
1.9984 
1.9970 

1-9955 
1.9941 
1.9926 

0.8948 
0.8947 
0.8945 
0.8944 
0.8943 
0.8942 
0.8940 
0.8939 
0.8938 

40 

0.4488 

0.5022 

1.9912 

0.8936 

4i 
42 
43 
44 
45 
46 

47 
48 
49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 

0.4491 

0.4493 
0.4496 

0.4498 
0.4501 
0.4504 
0.4506 
0.4509 
0.4511 

0.5020 
0.5029 
0.5033 

0.5037 
0.5040 
0.5044 
0.5048 
0.5051 
0.5055 

1.9897 
1.9883 
1.9868 
1.9854 
1.9840 
1.9825 
1.9811 
1.9797 
1.9782 

0.8935 
0.8934 
0.8932 
0.8931 
0.8930 
0.8928 
0.8927 
0.8926 
0.8925 

0.4514 

0.5059 

1.9768 

0.8923 

0.4517 
0.4519 
0.4522 
0.4524 
0.4527 
0.4530 
0.4532 
0.4535 
0.4537 

0.5062 
0.5066 
0.5070 

0.5073 
0.5077 
0.5081 

0.5084 
0.5088 
0.5092 

1-9754 
1.9740 

1-9725 
1.9711 
1.9697 
1.9683 
1.9669 
1.9654 
1.9640 

0.8922 
0.8921 
0.8919 
0.8918 
0.8917 
0.8915 
0.8914 
0.8913 
0.8911 

(JO 

0.4540 

0.5095 

1.9626 

0.8910 

Cos 

Cot 

Tan 

Sin 

' 

' 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 
4 

5 
6 

8 
9 
10 
ii 

12 
13 
14 

15 

16 

17 
18 

19 

20 

21 
22 
23 
24 

25 
26 

27 
28 
29 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 
42 

43 
44 
45 
46 

47 

48 

49 
50 

5i 

52 
53 

54 
55 
56 

57 
58 
59 

0.4540 

0.5095 

1.9626 

0.8910 

60 

59 

58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 

47 
46 
45 
44 

43 
42 

41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 

28 

27 
26 

25 
24 

23 

22 
21 
20 

J9 

18 

17 
16 
15 
14 
13 

12 
II 

10 

7 
6 
5 
4 
3 

2 

I 

0 

0.4542 
0-4545 
0.4548 
o.455o 
0-4553 
0.4555 
0.4558 
0.4561 
0.4563 

0.5099 
0.5103 
0.5106 
0.5110 

0.5114 
0.5117 

0.5121 
0.5125 
0.5128 

1.9612 
1.9598 
1.9584 

i«957o 
19556 
1.9542 

19528 

I.95I4 
1.9500 

0.8909 
0.8907 
0.8906 
0.8905* 
0.8903 
0.8902 
0.8901 
0.8899 
0.8898 

0.4566 

0.5132 

1.9486 

0.8897 

0.4568 
0.4571 
0.4574 
0.4576 

0-4579 
0.4581 

0.4584 
0.4586 
0.4589 

0.5136 
0.5139 
0.5143 

0.5147 
0.5150 

0.5154 
0.5158 
0.5161 
0.5165 

1.9472 
1.9458 
1.9444 

1.9430 
1.9416 
1.9402 
1.9388 
1-9375 
1.9361 

0.8895 
0.8894 
0.8893 
0.8892 
0.8890 
0.8889 
0.8888 
0.8886 
0.8885 

0.4592 

0.5169 

1-9347 

0.8884 

0.4594 
0.4597 
0-4599 
0.4602 
0.4605 
0.4607 
0.4610 
0.4612 
0.4615 

0.5172 
0.5176 
0.5180 

0.5184 
0.5187 
0.5191 

0.519! 
0.5198 
0.5202 

1-9333 
I.93I9 
1.9306 

1.9292 
1.9278 
1.9265 

1.9251 
1.9237 

1.9223 

0.8882 
0.8881 
0.8879 
0.8878 
0.8877 
0.8875 
0.8874 
0.8873 
0.8871 

0.4617 

0.5206 

1.9210 

0.8870 

0.4620 
0.4623 
0.4625 
0.4628 
0.4630 
0.4633 
0.4636 
0.4638 
0.4641 

0.5209 
0.5213 
0.5217 

0.5220 
0.5224 
0.5228 
0.5232 
0.5235 
0.5239 

1.9196 

1.9183 
1.9169 

I-9J55 
1.9142 
1.9128 
1.9115 
1.9101 
1.9088 

0.8869 
0.8867 
0.8866 
0.8865 
0.8863 
0.8862 
0.8861 
0.8859 
0.8858 

0.4643 

0.5243 

1.9074 

0.8857 

0.4646 
0.4648 
0.4651 
0.4654 
0.4656 
0.4659 
0.4661 
0.4664 
0.4666 

0.5246 
0.5250 
0.5254 
0.5258 
0.5261 
0.5265 
0.5269 
0.5272 
0.5276 

1.9061 
1.9047 
1.9034 
1.9020 
1.9007 
1.8993 
1.8980 
1.8967 
1-8953 

0.8855 
0.8854 
0.8853 
0.8851 
0.8850 
0.8849 
0.8847 
0.8846 
0.8844 

0.4669 

0.5280 

1.8940 

0.8843 

0.4672 
0.4674 
0.4677 
0.4679 
0.4682 
0.4684 
0.4687 
0.4690 
0.4692 

0.5284 
0.5287 
0.5291 

0.529! 
0.5298 
0.5302 
0.5306 
0.5310 
0.5313 

1.8927 
1.8913 
1.8900 

1.8887 
1.8873 
1.8860 
1.8847 
1.8834 
1.8820 

0.8842 
0.8840 
0.8839 
0.8838 
0.8836 
0.8835 
0.8834 
0.8832 
0.8831 

60 

0.4695 

0.5317 

1.8807 

0.8829 

Cos 

Cot 

Tan 

Sin 

; 

*153°  243°  *333° 


NATURAL 


*152°  242°  *332< 


122 

*118°  208°  *298° 


28< 


NATURAL 


29°      *H9°  209°  *299° 


' 

Sin 

Tan 

Cot 

Cos 

0 

0.4695 

0.5317 

1.8807 

0.8829 

60 

59 

58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 
47 
46 
45 
44 

43 
42 

4i 

40 

39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 

29 

28 

27 

26 
25 
24 
23 

22 
21 
20 
19 

18 
17 
16 
15 
M 
13 

12 
II 
10 

9 

8 

7 
6 
5 
4 
3 

2 

i 

0 

I 

•2. 

3 

4 
5 
6 

8 
9 
10 
ii 

12 
13 
14 
15 

16 

17 

18 

19 
20 

21 
22 
23 
24 

25 
26 

27 
28 

29 

30 

3i 

32 
33 
34 

35 
36 

37 
38 
39 
40 

4i 
42 
43 

44 
45 
46 

47 

48 

49 
50 

5i 

52 
53 
54 
55 
56 

57 
58 
59 
60 

0.4697 
0.4700 
0.4702 
0.4705 
0.4708 
0.4710 
0.4713 

0.4715 
0.4718 

0.5321 
0.5325 
0.5328 

0.5332 
0.5336 
0.5340 

0.5343 
0-5347 
o.535i 

1.8794 
1.8781 
1.8768 

1.8755 
1.8741 
1.8728 
1.8715 
1.8702 
1.8689 

0.8828 
0.8827 
0.8825 
0.8824 
0.8823 
0.8821 
0.8820 
0.8819 
0.8817 

0.4720 

0.5354 

1.8676 

0.8816 

0.4723 
0.4726 
0.4728 

0.4731 
0.4733 
0.4736 

0.4738 
0.4741 
0-4743 

0.5358 
0.5362 
0.5366 

0.5369 
0.5373 
0.5377 
0.5381 
0.5384 
0.5388 

1.8663 
1.8650 
1.8637 
1.8624 
1.8611 
1.8598 

1.8585 
1.8572 

1.8559 

0.8814 
0.8813 
0.8812 
0.8810 
0.8809 
0.8808 
0.8806 
0.8805 
0.8803 

0.4746 

0.5392 

1.8546 

0.8802 

0.4749 
0.4751 
0.4754 
0.4756 

0-4759 
0.4761 

0.4764 
0.4766 
0.4769 

0.5396 
0-5399 
0.5403 
0.5407 
0.5411 
0.5415 
0.5418 
0.5422 
0.5426 

1.8533 
1.8520 
1.8507 
1.8495 
1.8482 
1.8469 

1.8456 
1.8443 
1.8430 

0.8801 
0.8799 
0.8798 
0.8796 
0.8795 
0.8794 
0.8792 
0.8791 
0.8790 

0.4772 

o.543o 

1.8418 

0.8788 

0.4774 
0.4777 
0.4779 
0.4782 
0.4784 
0.4787 
0.4789 
0.4792 
0-4795 

0-5433 
0.5437 
0.5441 

0.5445 
0.5448 
0-5452 
0.5456 
0.5460 
0.5464 

1.8405 
1.8392 
1-8379 
1.8367 
1.8354 
1.8341 

1.8329 
1.8316 
1-8303 

0.8787 
0.8785 
0.8784 
0.8783 
0.8781 
0.8780 
0.8778 

0.8777 
0.8776 

0-4797 

0.5467 

1.8291 

0.8774 

0.4800 
0.4802 
0.4805 
0.4807 
0.4810 
0.4812 
0.4815 
0.4818 
0.4820 

o.547i 
0.5475 
0.5479 
0.5482 
0.5486 
0.5490 
0.5494 
0.5498 
0.5501 

1.8278 
1.8265 
1.8253 
1.8240 
1.8228 
1.8215 
1.8202 
1.8190 
1.8177 

0.8773 
0.8771 
0.8770 
0.8769 
0.8767 
0.8766 
0.8764 
0.8763 
0.8762 

0.4823 

0.5505 

1.8165 

0.8760 

0.4825 
0.4828 
0.4830 

0.4833 
0.4835 
0.4838 
0.4840 
0.4843 
0.4846 

0.5509 
0.5513 
0.5517 
0.5520 
0.5524 
0.5528 
0.5532 
0.5535 
0.5539 

1.8152 
1.8140 
1.8127 
1.8115 
1.8103 
1.8090 
1.8078 
1.8065 
1.8053 

0.8759 

0.8757 
0.8756 

0.875! 
0.8753 
0.8752 
0.8750 
0.8749 
0.8748 

0.4848 

0-5543 

1.8040 

0.8746 

Cos 

Cot 

Tan 

Sin 

' 

i 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 

6 

7 
8 

9 
10 
ii 

12 
13 
14 

15 

16 

17 

18 

*9 

20 

21 
22 
23 
24 
25 
26 

27 

28 
29 

30 

3i 
32 
33 
34 

35 
36 

37 
38 
39 
40 

4i 

42 

43 
44 
45 
46 

47 
48 
49 
50 

51 

52 
53 
54 
55 
56 

57 
58 
59 
60 

0.4848 

o.5543 

1.8040 

0.8746 

60 

59 

58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 
47 
46 
45 
44 
43 
42 
4i 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 

28 

27 
26 

25 
24 

23 

22 
21 

0.4851 

0.4853 
0.4856 

0.4858 
0.4861 
0.4863 
0.4866 
0.4868 
0.4871 

0-5547 
0.5551 
0.5555 
0.5558 
0.5562 
0.5566 

0.5570 
0-5574 
o.5577 

1.8028 
i.  8016 
1.8003 
1.7991 

1-7979 
1.7966 

1-7954 
1.7942 
1.7930 

0.8745 
0.8743 
0.8742 
0.8741 
0.8739 
0.8738 
0.8736 

0.8735 
0.8733 

0.4874 

0.5581 

1.7917 

0.8732 

0.4876 
0.4879 
0.4881 
0.4884 
0.4886 
0.4889 
0.4891 
0.4894 
0.4896 

0.5585 
0.5589 
0-5593 

0.5596 
0.5600 
0.5604 
0.5608 
0.5612 
0.5616 

1.7905 
1-7893 
1.7881 

1.7868 
1.7856 
1.7844 

1.7832 
1.7820 
1.7808 

0.8731 
0.8729 
0.8728 
0.8726 
0.8725 
0.8724 
0.8722 
0.8721 
0.8719 

0.4899 

0.5619 

1.7796 

0.8718 

0.4901 
0.4904 
0.4907 

0.4909 
0.4912 
0.4914 
0.4917 
0.4919 
0.4922 

0.5623 
0.5627 
0.5631 

0.563! 
0.5639 
0.5642 

0.5646 
0.5650 

0.5654 

1.7783 
1.7771 
1-7759 
1.7747 
1.7735 
1.7723 

1.7711 
1.7699 

1.7687 

0.8716 
0.8715 
0.8714 
0.8712 
0.8711 
0.8709 
0.8708 
0.8706 
0.8705 

0.4924 

0.5658 

1.7675 

0.8704 

0.4927 
0.4929 
0.4932 

0-4934 
0.4937 
0.4939 
0.4942 
0.4944 
0.4947 

0.5662 
0.5665 
0.5669 

0.5673 

0.5677 
0.5681 

0.5685 
0.5688 
0.5692 

1.7663 
1.7651 
1.7639 
1.7627 
1.7615 
1.7603 

I.759I 
1.7579 
1.7567 

0.8702 
0.8701 
0.8699 
0.8698 
0.8696 
0.8695 
0.8694 
0.8692 
0.8691 

0.4950 

0.5696 

1.7556 

0.8689 

20 

0.4952 
0-4955 
0.4957 
0.4960 
0.4962 
0.4965 
0.4967 
0.4970 
0.4972 

0.5700 
0.5704 
0.5708 

0.5712 
0.5715 
0.5719 
0.5723 
0.5727 
0.5731 

1-7544 
1.7532 
1.7520 
1.7508 
1.7496 
1.748! 

1-7473 
1.7461 

1-7449 

0.8688 
0.8686 
0.8685 
0.8683 
0.8682 
0.8681 
0.8679 
0.8678 
0.8676 

19 

18 

17 

16 
15 
M 
13 

12 
II 

0-4975 

0.5735 

1-7437 

0.8675 

10 

0.4977 
0.4980 
0.4982 
0.4985 
0.4987 
0.4990 
0.4992 
0.4995 
0.4997 

0.5739 
0.5743 
0.5746 
0.5750 
0.5754 
0.5758 
0.5762 
0.5766 
0.5770 

1.7426 
1.7414 
1.7402 

I-739I 
1-7379 
1.7367 
1-7355 
1-7344 
1-7332 

0.8673 
0.8672 
0.8670 
0.8669 
0.8668 
0.8666 
0.8665 
0.8663 
0.8662 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0.5000 

0.5774 

1.7321 

0.8660 

0 

Cos 

Cot 

Tan 

Sin 

' 

*151°  241°  *331°       61' 


NATURAL 


60°      *150°  240°  *330° 


120°  210°  *300° 


NATURAL 


sr 


123 
*121°  211°  *301° 


Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 
5 
6 

8 
9 
10 
ii 

12 
13 
14 
15 

16 

17 

18 

19 
20 

21 
22 
23 
24 

25 
26 

27 
28 
29 

SO 

31 

32 

33 
34 
35 
36 

37 
38 
39 
40 

0.5000 

0-5774 

1.7321 

0.8660 

60 

59 

58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 

48 
47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 
28 

27 
26 
25 
24 
23 

22 
21 

20 

J9 

18 

17 
16 
15 
14 

13 

12 
II 

10 

9 

8 

7 
6 
5 
4 

3 

2 

I 

0 

0.5003 
0.5005 
0.5008 
0.5010 
0.5013 
0.5015 
0.5018 
0.5020 
0.5023 

o.5777 
0.5781 

0.5785 
0.5789 
0-5793 
o.5797 
0.5801 
0.5805 
0.5808 

1.7309 
1.7297 
1.7286 
1.7274 
1.7262 
1.7251 
1.7239 
1.7228 
1.7216 

0.8659 
0.8657 
0.8656 
0.8654 
0.8653 
0.8652 
0.8650 
0.8649 
0.8647 

0.5025 

0.5812 

1.7205 

0.8646 

0.5028 
0.5030 
0.5033 

0.5035 
0.5038 
0.5040 

0.5043 
0.5045 
0.5048 

0.5816 
0.5820 
0.5824 
0.5828 
0.5832 
0.5836 
0.5840 
0.5844 
0.5847 

1-7193 
1.7182 
1.7170 

1-7159 
1.7147 
1.7136 
1.7124 

1.7113 
1.7102 

0.8644 
0.8643 
0.8641 
0.8640 
0.8638 
0.8637 
0.8635 
0.8634 
0.8632 

0.5050 

0.5851 

1.7090 

0.8631 

0.5053 
0.5055 
0.5058 

0.5060 
0.5063 
0.5065 
0.5068 
0.5070 
0.5073 

0.5855 
0.5859 
0.5863 

0.5867 
0.5871 
0.5875 
0.5879 
0.5883 
0.5887 

1.7079 
1.7067 
1.7056 

1.7045 
1-7033 
1.7022 

1.7011 
1.6999 

1.6988 

^.8630 
0.8628 
0.8627 
0.8625 
0.8624 
0.8622 
0.8621 
0.8619 
0.8618 

0.5075 

0.5890 

1.6977 

0.8616 

0.5078 
0.5080 
0.5083 
0.5085 
0.5088 
0.5090 

0.5093 
0.5095 
0.5098 

0.5894 
0.5898 
0.5902 
0.5906 
0.5910 
0.5914 
0.5918 
0.5922 
0.5926 

1.6965 
1.6954 
1.6943 
1.6932 
1.6920 
1.6909 
1.6898 
1.6887 
1.6875 

0.8615 
0.8613 
0.8612 
0.8610 
0.8609 
0.8607 
0.8606 
0.8604 
0.8603 

0.5100 

0.5930 

1.6864 

0.8601 

4i 

42 

43 
44 
45 
46 

47 
48 
49 
50 

0.5103 
0.5105 
0.5108 
0.5110 
0.5113 
0.5115 
0.5118 
0.5120 
0.5123 

0-5934 
0.5938 
0.5942 

0-5945 
0.5949 

0-5953 
0-5957 
0.5961 
0.5965 

1.6853 
1.6842 
1.6831 
1.6820 
1.6808 
1.6797 
1.6786 

1.6775 
1.6764 

0.8600 
0.8599 
0.8597 
0.8596 
0.8594 
0.8593 
0.8591 
0.8590 
0.8588 

0.5125 

0.5969 

1.6753 

0.8587 

5i 
52 
53 
54 
55 
56 

57 
58 
59 

0.5128 
0.5130 
0.5133 
0.5135 
0.5138 
0.5140 

0.5143 
0.5145 
0.5148 

o.5973 
0-5977 
0.5981 

0.5985 
0.5989 
0-5993 

0-5997 
0.600  1 
0.6005 

1.6742 
1.6731 
1.6720 
1.6709 
1.6698 
1.6687 
1.6676 
1.6665 
1.6654 

0.8585 
0.8584 
0.8582 
0.8581 

0.8579 
0.8578 

0.8576 
0.8575 
0.8573 

60 

0.5150 

0.6009 

1.6643 

0.8572 

Cos 

Cot 

Tan 

Sin 

' 

*149°  239°  *329° 


NATURAL 


' 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 

4 

6 

8 
9 
10 
ii 

12 
13 
14 
15 

16 

17 
18 

*9 

20 

21 
22 
23 
24 

25 
26 

27 

28 

29 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 
42 

43 
44 
45 
46 

47 
48 

49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 
60 

0.5150 

0.6009 

1.6643 

0.8572 

60 

59 

58 
57 
56 
55 
54 
53 
52 
5i 
50 

49 

48 

47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 

22 
21 
20 
J9 

18 
17 
16 
15 
14 
13 

12 
II 

10 

9 
8 

7 
6 
5 
4 
3 

2 

I 

0 

0.5153 
o.5i55 
0.5158 
0.5160 
0.5163 
0.5165 
0.5168 
0.5170 
0-5173 

0.6013 
0.6017 
0.6020 
0.6024 
0.6028 
0.6032 
0.6036 
0.6040 
0.6044 

1.6632 
1.6621 
1.6610 
1.6599 
1.6588 
1.6577 
1.6566 
1.6555 
1.6545 

0.8570 
0.8569 
0.8567 
0.8566 
0.8564 
0.8563 
0.8561 
0.8560 
0.8558 

0.5175 

0.6048 

1.6534 

0.8557 

0.5178 
0.5180 
0.5183 
0.5185 
0.5188 
0.5190 

0.5193 
0.5195 
0.5198 

0.6052 
0.6056 
0.6060 
0.6064 
0.6068 
0.6072 
0.6076 
0.6080 
0.6084 

1.6523 
1.6512 
1.6501 
1.6490 
1.6479 
1.6469 

1.6458 
1.6447 
1.6436 

0.8555 
0.8554 
0.8552 
0.8551 
0.8549 
0.8548 
0.8546 
0.8545 
0.8543 

0.5200 

0.6088 

1.6426 

0.8542 

0.5203 
0.5205 
0.5208 
0.5210 
0.5213 
0.5215 
0.5218 
0.5220 
0.5223 

0.6092 
0.6096 
0.6100 
0.6104 
0.6108 
0.6112 
0.6116 
0.6120 
0.6124 

1.6415 
1.6404 

1.6393 
1.6383 
1.6372 
1.6361 

1.6351 
1.6340 
1.6329 

0.8540 

0.8539 
0.8537 

0.8536 

0.8534 
0.8532 

0.8531 
0.8529 

0.8528 

Q.-5225 

0.6128 

1.6319 

0.8526 

0.5227 
0.5230 
0.5232 

0.523! 
0.5237 
0.5240 

0.5242 

0.524! 
0.5247 

0.6132 
0.6136 
0.6140 
0.6144 
0.6148 
0.6152 
0.6156 
0.6160 
0.6164 

1.6308 
1.6297 
1.6287 
1.6276 
1.6265 
1.6255 
1.6244 
1.6234 
1.6223 

0.8525 
0.8523 
0.8522 
0.8520 
0.8519 
0.8517 
0.8516 
0.8514 
0.8513 

0.5250 

0.6168 

1.6212 

0.8511 

0.5252 
0.5255 
0.5257 
0.5260 
0.5262 
0.5265 
0.5267 
0.5270 
0.5272 

0.6172 
0.6176 
0.6180 
0.6184 
0.6188 
0.6192 
0.6196 
0.6200 
0.6204 

1.6202 
1.6191 
1.6181 
1.6170 
i.  6160 
1.6149 

1.6139 
1.6128 
1.6118 

0.8510 
0.8508 
0.8507 
0.8505 
0.8504 
0.8502 
0.8500 
0.8499 
0.8497 

0.527! 

0.6208 

1.6107 

0.8496 

0.5277 
0-S279 
0.5282 

0.5284 
0.5287 
0.5289 
0.5292 

0.5294 
0.5297 

0.6212 
0.6216 
0.6220 
0.6224 
0.6228 
0.6233 
0.6237 
0.6241 
0.6245 

1.6097 
1.6087 
1.6076 
1.6066 
1.6055 
1.6045 
1.6034 
1.6024 
1.6014 

0.8494 
0.8493 
0.8491 
0.8490 
0.8488 
0.8487 
0.8485 
0.8484 
0.8482 

0.5299 

0.6249 

1.6003 

0.8480 

Cos 

Cot 

Tan 

Sin 

' 

58' 


*148°  238°  *328° 


124 

*122°  212°  *302° 


32° 


NATURAL 


33°      *123°  213°  *303° 


' 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 

4 
5 
6 

8 
9 
10 
ii 

12 
13 
14 
15 

16 

17 
18 

19 
20 

21 
22 
23 
24 
25 
26 

27 

28 

29 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 
42 
43 
44 
45 
46 

47 

48 

49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 
60 

0.5299 

0.6249 

1.6003 

0.8480 

60 

59 
58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 

48 

47 
46 
45 
44 

43 
42 

4i 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 

28 

27 
26 
25 
24 
23 

22 
21 
20 
19 

18 
17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.5302 
0.5304 
0.5307 
0.5309 
0.5312 
0.5314 
0.5316 

0.5319 
0.5321 

0.6253 
0.6257 
0.6261 
0.6265 
0.6269 
0.6273 
0.6277 
0.6281 
0.6285 

1-5993 
1.5983 
1-5972 
1.5962 
1-5952 
1.5941 

i.593i 
1.5921 
1.5911 

0.8479 

0.8477 
0.8476 

0.8474 
0.8473 
0.8471 
0.8470 
0.8468 
0.8467 

0.5324 

0.6289 

1.5900 

0.8465 

0.5326 
0.5329 
o-533i 
0.5334 
0.5336 
0-5339 
o.534i 
0-5344 
0.5346 

0.6293 
0.6297 
0.6301 
0.6305 
0.6310 
0.6314 
0.6318 
0.6322 
0.6326 

1.5890 

1.5880 
1.5869 

1-5859 
1.5849 

i.  ,5  839 
i.  '5  829 

1.5818 
1.5808 

0.8463 
0.8462 
0.8460 
0.8459 
0.8457 
0.8456 
0.8454 

0.8453 
0.8451 

0.5348 

0.6330 

1.5798 

0.8450 

0-5351 
0-5353 
0.5356 

0.5358 
0.5361 
0.5363 
0.5366 
0.5368 
o.537i 

0.6334 
0.6338 
0.6342 
0.6346 
0.6350 
0.6354 
0.6358 
0.6363 
0.6367 

1.5788 
1.5778 
1.5768 

1.5757 
1-5747 
1-5737 
1.5727 
I-57I7 
1.5707 

0.8448 
0.8446 
0.8445 
0.8443 
0.8442 
0.8440 
0.8439 
0.8437 
0.8435 

0-5373 

0.6371 

1-5697 

0.8434 

0-5375 
0.5378 
0.5380 

0.5383 
0.5385 
0.5388 

0.5390 
0-5393 
0-5395 

0.6375 
0.6379 
0.6383 
0.6387 
0.6391 
0.6395 
0.6399 
0.6403 
0.6408 

1.5687 

1-5677 
1.5667 

1.5657 
1.5647 
1.5637 
1.5627 
1.5617 
1.5607 

0.8432 
0.8431 
0.8429 
0.8428 
0.8426 
0.8425 
0.8423 
0.8421 
0.8420 

0.5398 

0.6412 

1-5597 

0.8418 

0.5400 
0.5402 
0.5405 

0.5407 
0.5410 
0.5412 

0.541! 
o.54i7 
0.5420 

0.6416 
0.6420 
0.6424 
0.6428 
0.6432 
0.6436 
0.6440 

0.6445 
0.6449 

1.5587 
1-5577 
1.5567 
1-5557 
1-5547 
1-5537 
1.5527 
I-55I7 
1.5507 

0.8417 
0.8415 
0.8414 
0.8412 
0.8410 
0.8409 
0.8407 
0.8406 
0.8404 

0.5422 

0.6453 

1-5497 

0.8403 

0.5424 

0.5427 
0.5429 

0.5432 
0-5434 
0.5437 

0.5439 
0.5442 

0-5444 

0.6457 
0.6461 
0.6465 
0.6469 
0.6473 
0.6478 
0.6482 
0.6486 
0.6490 

1.5487 
1-5477 
1.5468 

1.5458 
1.5448 
1.5438 
1.5428 
1.5418 
1.5408 

0.8401 
0.8399 
0.8398 
0.8396 
0.8395 
0.8393 
0.8391 
0.8390 
0.8388 

0.5446 

0.6494 

1-5399 

0.8387 

Cos 

Cot 

Tan 

Sin 

' 

•147"  237°  *327° 


57' 


NATURAL 


' 

Sin 

Tan 

Cot 

Cos 

0 

0.5446 

0.6494 

-5399 

0.8387 

60 

I 

2 

3 

4 

6 

7 
8 

9 

0-5449 
o.545i 
0.5454 
0.5456 

0-5459 
0.5461 

0.5463 
0.5466 
0.5468 

0.6498 
0.6502 
0.6506 
0.6511 
0.6515 
0.6519 
0.6523 
0.6527 
0.6531 

.5389 
•5379 
•5369 

•5359 
.5350 
•  5340 
•5330 
•5320 
•5311 

0.8385 
0.8384 
0.8382 
0.8380 
0.8379 
0.8377 
0.8376 
0.8374 
0.8372 

59 

58 
57 
56 
55 
54 

53 
52 
5i 

10 

o.547i 

0.6536 

•5301 

0.8371 

50 

ii 

12 
13 
14 
15 

16 

17 

18 

19 

0-5473 
0.5476 
0.5478 
0.5480 
0.5483 
0.5485 
0.5488 
0.5490 
0-5493 

0.6540 
0.6544 
0.6548 
0.6552 
0.6556 
0.6560 
0.6565 
0.6569 
0.6573 

.5291 
.5282 
.5272 
.5262 
-5253 
.5243 

.5233 
.5224 
•5214 

0.8369 
0.8368 
0.8366 
0.8364 
0.8363 
0.8361 
0.8360 
0.8358 
0.8356 

49 
48 
47 
46 
45 
44 

43 
42 
4i 

20 

0-5495 

0.6577 

5204 

0.835! 

40 

21 
22 
23 
24 
25 
26 

27 

28 

29 

0.5498* 
0.5500 
0.5502 

0.5505 
0.5507 
0.5510 

0.5512 
o.55i5 
0.5517 

0.6581 
0.6585 
0.6590 
0.6594 
0.6598 
0.6602 
0.6606 
0.6610 
0.6615 

.5195 
•5185 
•5175 
.5166 
.5156 
.5147 

.5137 
.5127 

.5118 

0.8353 
0.8352 
0.8350 
0.8348 
0.8347 
0.8345 
0.8344 
0.8342 
0.8340 

39 
38 
37 
36 
35 
34 

33 
32 
3i 

30 

0.5519 

0.6619 

.5108 

0.8339 

30 

3i 

32 
33 
34 
35 
36 

37 

38 
39 

0.5522 
0.5524 
0.5527 
0.5529 
o.553i 
0.5534 
0.5536 
0.5539 
o.554i 

0.6623 
0.6627 
0.6631 
0.6636 
0.6640 
0.6644 
0.6648 
0.6652 
0.6657 

•5099 
.5089 
.5080 

•5070 
.5061 
-5051 
.5042 
.5032 
•  5023 

0.8337 
0.8336 
0.8334 
0.8332 
0.8331 
0.8329 
0.8328 
0.8326 
0.8324 

29 
28 
27 
26 
25 
24 
23 

22 
21 

40 

0-5544 

0.6661 

•5013 

0.8323 

20 

4i 
42 
43 
44 
45 
46 

47 

48 
49 

0.5546 
0.5548 
o.555i 

0.5553 
0.5556 
0.5558 
0.5561 
0.5563 
0.5565 

0.6665 
0.6669 
0.6673 
0.6678 
0.6682 
0.6686 
0.6690 
0.6694 
0.6699 

.5004 
.4994 
4985 

-4975 
.4966 

•4957 

-4947 
.4938 
.4928 

0.8321 
0.8320 
0.8318 
0.8316 
0.8315 
0.8313 
0.8311 
0.8310 
0.8308 

J9 

18 
17 
16 
15 
14 
13 

12 
II 

60 

0.5568 

0.6703 

.4919 

0.8307 

10 

5i 
52 
53 
54 
55 
56 

57 
58 
59 

o.557o 
0.5573 
0.5575 

0-5577 
0.5580 
0.5582 

0.5585 
0.5587 
0.5590 

0.6707 
0.6711 
0.6715 
0.6720 
0.6724 
0.6728 
0.6732 
0.6737 
0.6741 

.4910 
1.4900 
1.4891 
1.4882 
1.4872 
1.4863 
1.4854 
1.4844 
1.483! 

0.8305 
0.8303 
0.8302 
0.8300 
0.8298 
0.8297 
0.8295 
0.8294 
0.8292 

9 

8 

7 
6 
5 
4 
3 

2 

I 

60 

C.^02 

0.6745 

1.4820 

0.8290 

0 

Cos 

Cot 

Tan 

Sin 

' 

56' 


*H6°  236°  *326° 


*124°    214°  *304° 


NATURAL 


125 
35°    *125°    215°  *305° 


' 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 

6 

8 
9 
10 
ii 

12 
13 
14 
15 
16 

17 

18 
19 
20 

21 
22 
23 
24 

25 
26 

27 
28 

29 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 

42 

43 
44 
45 
46 

47 

48 

49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 
60 

0.5592 

0.6745 

1.4826 

0.8290 

60 

59 

58 
57 
56 
55 
54 

53 
52 
51 
50 

49 
48 

47 
46 
45 
44 

43 
42 
4i 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 
29 
28 
27 
26 

25 
24 

23 

22 
21 

20 

19 

18 
17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 
I 

0 

0.5594 
0.5597 
0.5599 
0.5602 
0.5604 
0.5606 
0.5609 
0.5611 
0.5614 

0.6749 
0.6754 
0.6758 
0.6762 
0.6766 
0.6771 
0.6775 

0.6779 
0.6783 

1.4816 
1.4807 
1.4798 

1.4788 

1-4779 
1.4770 

1.4761 

i.475i 
1.4742 

0.8289 
0.8287 
0.8285 
0.8284 
0.8282 
0.8281 
0.8279 
0.8277 
0.8276 

0.5616 

0.6787 

1-4733 

0.8274 

0.5618 
0.5621 
0.5623 
0.5626 
0.5628 
0.5630 

0.5633 
0.5635 
0.5638 

0.6792 
0.6796 
0.6800 
0.6805 
0.6809 
0.6813 
0.6817 
0.6822 
0.6826 

.  1-4724 
M7I5 
14705 
1.4696 
1.4687 
1.4678 
1.4669 
1.4659 
1.4650 

0.8272 
0.8271 
0.8269 
0.8268 
0.8266 
0.8264 
0.8263 
0.8261 
0.8259 

0.5640 

0.6830 

1.4641 

0.8258 

0.5642 

0.5645 
0.5647 

0.5650 
0.5652 
0.5654 
0.5657 
0.5659 
0.5662 

0.6834 
0.6839 
0.6843 
0.6847 
0.6851 
0.6856 
0.6860 
0.6864 
0.6869 

1.4632 
1-4623 
1.4614 

1.4605 
1.4596 
1.4586 

1-4577 
1.4568 

1-4559 

0:8256 
0.8254 
0.8253 
0.8251 
0.8249 
0.8248 
0.8246 
0.8245 
0.8243 

0.5664 

0.6873 

1-4550 

0.8241 

0.5666 
0.5669 
0.5671 

0.5674 
0.5676 
0.5678 
0.5681 
0.5683 
0.5686 

0.6877 
0.6881 
0.6886 
0.6890 
0.6894 
0.6899 
0.6903 
0.6907 
0.6911 

I-454I 
1.4532 
1.4523 
I.45I4 
1.4505 
1.4496 
1.4487 
1.4478 
1.4469 

0.8240 
0.8238 
0.8236 
0.8235 
0.8233 
0.8231 
0.8230 
o  8228 
0.8226 

0.5688 

0.6916 

1.4460 

0.8225 

0.5690 
0.5693 
0.5695 
0.5698 
0.5700 
0.5702 

0.5705 
0.5707 
0.5710 

0.6920 
0.6924 
0.6929 
0.6933 
0.6937 
0.6942 
0.6946 
0.6950 
0.6954 

I-445I 
1.4442 

1-4433 
1.4424 

I.44I5 
1.4406 

1.4397 
1.4388 

1.4379 

0.8223 
0.8221 
0.8220 
0.8218 
0.8216 
0.8215 
0.8213 
0.8211 
0.8210 

0.5712 

0.6959 

1.4370 

0.8208 

o.57i4 
0.5717 
0.5719 
0.5721 
0.5724 
0.5726 

0.5729 
o.573i 
0-5733 

0.6963 
0.6967 
0.6972 
0.6976 
0.6980 
0.6985 
0.6989 
0.6993 
0.6998 

1.4361 
1.4352 
1-4344 

1-4335 
1.4326 

I.43I7 
1.4308 
1.4299 
1.4290 

0.8207 
0.8205 
0.8203 
0.8202 
0.8200 
0.8198 
0.8197 
0.8195 
0.8193 

0.5736 

0.7002 

1.4281 

0.8192 

Cos 

Cot 

Tan 

Sin 

' 

f 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 
4 
5 
6 

7 
8 

9 
10 
ii 

12 
13 
14 

15 

16 
i? 

id 

19 

20 

21 
22 
23 
24 

25 
26 

27 

28 

29 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 

42 
43 
44 
45 
46 

47 

48 

49 
50 

5i 

52 
53 
54 
55 
56 

57 
58 
59 
60 

0.5736 

0.7002 

1.4281 

0.8192 

60 

59 
58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 

48 

47 
46 
45 
44 

43 
42 
4i 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 
28 

27 
26 
25 
24 
23 

22 
21 
20 
19 

18 
17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.5738 
0.5741 
0-5743 

0-5745 
0.5748 
0.5750 
0.5752 
0-5755 
0-5757 

0.7006 
0.7011 
0.7015 
0.7019 
0.7024 
0.7028 
0.7032 
0.7037 
0.7041 

1-4273 
1.4264 

14255 
1.4246 

1.4237 
1.4229 

1.4220 
1.4211 
1.4202 

0.8190 
0.8188 
0.8187 
0.8185 
0.8183 
0.8181 
0.8  1  80 
0.8178 
0.8176 

0.5760 

0.7046 

1.4193 

0.8175 

0.5762 
0.5764 
0.5767 
0.5769 
o.577i 
0.5774 
0.5776 

0.5779 
0.5781 

0.7050 
0.7054 
0.7059 

0.7063 
0.7067 
0.7072 
0.7076 
0.7080 
0.7085 

1.4185 
1.4176 
1.4167 
1.4158 
1.4150 
1.4141 

1.4132 
1.4124 
1-4115 

0.8173 
0.8171 
0.8170 
0.8168 
0.8  1  66 
0.8165 
0.8163 
0.8161 
0.8  1  60 

0.5783 

0.7089 

1.4106 

0.8158 

0.5786 
0.5788 
0.5790 
0-5793 
0-5795 
0.5798 
0.5800 
0.5802 
0.5805 

0.7094 
0.7098 
0.7102 
0.7107 
0.7111 
0.7115 
0.7120 
0.7124 
0.7129 

1.4097 
1.4089 
1.4080 
1.4071 
^1.4063 
1.4054 

1.4045 

1.4037 
1.4028 

0.8156 
0.8155 
0.8153 
0.8151 
0.8150 
0.8148 
0.8146 
0.8145 
0.8143 

0.5807 

0.7133 

1.4019 

0.8141 

0.5809 
0.5812 
0.5814 
0.5816 
0.5819 
0.5821 
0.5824 
0.5826 
0.^5828 

0.7137 
0.7142 
0.7146 
0.7151 
0.7155 
0.7159 
0.7164 
0.7168 
0.7173 

1.4011 
1.4002 
1-3994 

1-3985 
1.3976 
1.3968 

1-3959 
i.395i 
1-3942 

0.8139 
0.8138 
0.8136 
0.8134 
0.8133 
0.8131 
0.8129 
0.8128 
0.8126 

0.5831 

0.7177 

1-3934 

0.8124 

0.5833 
0.5835 
0.5838 

0.5840 
0.5842 
0.584! 

0.5847 
0.5850 
0.5852 

0.7181 
0.7186 
0.7190 

0.719! 
0.7199 
0.7203 
0.7208 
0.7212 
0.7217 

1.3925 
1.3916 
1.3908 
1.3899 
1.3891 
1.3882 

1.3874 
1.3865 
1.3857 

0.8123 
0.8121 
0.8119 
0.8117 
0.8116 
0.8114 
0.8112 
0.8111 
0.8109 

0.5854 

0.7221 

1.3848 

0.8107 

0.5857 
0.5859 
0.5861 

0.5864 
0.5866 
0.5868 

0.5871 
0.5873 
0.5875 

0.7226 
0.7230 
0.7234 
0.7239 

0.7243 
0.7248 

0.7252 
0.7257 
0.7261 

1.3840 
1.3831 
1.3823 

1.3814 
1.3806 
1.3798 

1.3789 
1.3781 
1.3772 

0.8106 
0.8104 
0.8102 
0.8100 
0.8099 
0.8097 
0.8095 
0.8094 
0.8092 

0.5878 

0.7265 

1.3764 

0.8090 

Cos 

Cot 

Tan 

Sin 

' 

*145°  235°  *325 


NATURAL 


'144°  234°  *324° 


126 

*126°  216°  *306° 


36< 


NATURAL 


37' 


*127°  217°  *307° 


' 

Sin 

Tan 

Cot 

Cos 

0 

0.5878 

0.7265 

1-3764 

0.8090 

60 

I 

2 

3 
4 
5 
6 

8 
9 

0.5880 
0.5883 
0.5885 
0.5887 
0.5890 
0.5892 
0.5894 
0.5897 
0.5899 

0.7270 
0.7274 
0.7279 
0.7283 
0.7288 
0.7292 

0-7297 
0.7301 
0.7306 

1.3755 
1-3747 
1-3739 
1-3730 
1.3722 
I.37I3 
1.3705 
1.3697 
1.3688 

0.8088 
0.8087 
0.8085 
0.8083 
0.8082 
0.8080 
0.8078 
0.8076 
0.8075 

59 

58 
57 
56 
55 
54 

53 
52 
5i 

10 

0.5901 

0.7310 

1.3680 

0.8073 

50 

ii 

12 
13 
14 
15 

16 

17 
18 

!9 

0.5904 
0.5906 
0.5908 
0.5911 
o-59!3 
0.5915 
0.5918 
0.5920 
0.5922 

0.7314 
0-7319 
0.7323 
0.7328 
0-7332 
0-7337 

0-7341 
0.7346 
0.7350 

1-3672 
1-3663 
1.3655 
1-3647 
1-3638 
1-3630 
1.3622 

1-3613 
1.3605 

0.8071 
0.8070 
0.8068 
0.8066 
0.8064 
0.8063 
0.8061 
0.8059 
0.8058 

49 
48 
47 
46 
45 
44 

43 
42 
4i 

20 

0.5925 

0-7355 

1-3597 

0.8056 

40 

21 
22 
23 
24 
25 
26 

27 

28 
29 

0.5927 
0.5930 
0.5932 
0-5934 
0-5937 
0.5939 
0.5941 
0.5944 
0.5946 

0.7359 
0.7364 
0.7368 

0-7373 
0.7377 
0.7382 

0.7386 
0.7391 
0.7395 

1.3588 
i.358o 
1-3572 
1-3564 
1-3555 
1-3547 

1-3539 
I-353I 
1.3522 

0.8054 
0.8052 
0.8051 
0.8049 
0.8047 
0.8045 
0.8044 
0.8042 
0.8040 

39 
38 
37 
36 
35 
34 

33 
32 
3i 

30 

0.5948 

0.7400 

I-35I4 

0.8039 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 

o.595i 
0-5953 
0-5955 
0.5958 
0.5960 
0.5962 

0.596! 
0.5967 

0.5969 

0.7404 
0.7409 
0.7413 
0.7418 
0.7422 
0.7427 

o.743i 
0.7436 
0.7440 

1.3506 
1.3498 
1.3490 

i.348i 
1-3473 
1.3465 
1-3457 
1-3449 
1.3440 

0.8037 
0.8035 
0.8033 
0.8032 
0.8030 
0.8028 
0.8026 
0.8025 
0.8023 

29 
28 
27 
26 

25 
24 

23 

22 
21 

40 

0-5972 

0.7445 

1-3432 

0.8021 

20 

4i 
42 
43 
44 
45 
46 

47 
48 
49 

0-5974 
0.5976 
0-5979 
0.5981 

0.5983 
0.5986 

0.5988 
0.5990 
0-5993 

0.7449 
0-7454 
0.7458 

0-7463 
0.7467 
0.7472 

0.7476 
0.7481 
0.7485 

1.3424 
1-3416 
1-3408 

1-3400 
1.3392 
1.3384 
1-3375 
1.3367 
1.3359 

0.8019 
0.801  8 
0.8016 
0.8014 
0.8013 
0.8011 
0.8009 
0.8007 
0.8006 

19 

18 
17 
16 
15 
14 
13 

12 
II 

50 

0-5995 

0.7490 

I.335I 

0.8004 

10 

51 
52 
53 
54 
55 
56 

57 
58 
59 

0-5997 
0.6000 
0.6002 
0.6004 
0.6007 
0.6009 
0.6011 
0.6014 
0.60  1  6 

0-7495 
0-7499 
0.7504 

0.7508 
0.7513 
0.7517 
0.7522 
0.7526 
o.753i 

1.3343 
1.3335 
1.3327 

I.33I9 
I-33II 
I.3303 

1.329! 
1.3287 
1-3278 

0.8002 
0.8000 
0-7999 
0-7997 
0-7995 
0-7993 
0.7992 
0.7990 
0.7988 

9 

8 

7 
6 
5 
4 
3 

2 

I 

60 

0.6018 

0.7536 

1.3270 

0.7986 

0 

Cos 

|  Cot 

Tan 

Sin 

' 

i 

Sin 

Tan 

Cot 

Cos 

0 

0.601  8 

0.7536 

1.3270 

0.7986 

60 

I 

2 

3 
4 

6 

7 
8 

9 

0.6020 
0.6023 
0.6025 
0.6027 
0.6030 
0.6032 
0.6034 
0.6037 
0.6039 

0.7540 
0-7545 
0-7549 

0-7554 
0.7558 
0.7563 
0.7568 
0.7572 
0-7577 

1.3262 
1.3254 
1-3246 

1.3238 
1.3230 
1.3222 

1.3214 
1.3206 
1.3198 

0-7985 
0.7983 
0.7981 

0-7979 
0.7978 

0.7976 

0-7974 
0.7972 

o.797i 

59 

58 
57 
56 
55 
54 

53 
52 

5i 

10 

0.6041 

0.7581 

1.3190 

0.7969 

50 

ii 

12  ' 
13 
14 
15 

16 

17 

18 

19 

0.6044 
0.6046 
0.6048 
0.6051 
0.6053 
0.6055 
0.6058 
0.6060 
0.6062 

0.7586 
0.759° 
0.759! 
0.7600 
0.7604 
0.7609 
0.7613 
0.7618 
0.7623 

1.3182 

i.3i75 
1.3167 

1-3159 
1-3151 
i.3H3 

1-3135 
1.3127 
1.3119 

0.7967 
0.7965 
0.7964 
0.7962 
0.7960 
0.7958 
0.7956 
0-7955 
0-7953 

49 

48 

47 
46 
45 
44 

43 
42 
4i 

20 

0.6065 

0.7627 

1.3111 

0-7951 

40 

21 
22 
23 
24 

25 
26 

27 
28 
29 

0.6067 
0.6069 
0.6071 
0.6074 
0.6076 
0.6078 
0.608  1 
0.6083 
0.6085 

0.7632 
0.7636 
0.7641 
0.7646 
0.7650 
0.7655 

0.7659 
0.7664 
0.7669 

1-3103 

1.3095 
1-3087 

1-3079 
1-3072 
1.3064 

1-3056 
1.3048 
1.3040 

0-7949 
0.7948 
0.7946 

0-7944 
0.7942 

0-7941 
0.7939 
0-7937 
0-7935 

39 
38 
37 
36 
35 
34 

33 
32 
31 

30 

0.6088 

0.7673 

1.3032 

0-7934 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 

0.6090 
0.6092 
0.6095 
0.6097 
0.6099 
0.6101 
0.6104 
0.6106 
0.6108 

0.7678 
0.7683 
0.7687 
0.7692 
0.7696 
0.7701 
0.7706 
0.7710 
0.7715 

1.3024 
1.3017 
1-3009 
1.3001 

1.2993 
1.2985 

1.2977 
1.2970 
1.2962 

0.7932 
0.7930 
0.7928 

0.7926 
0.7925 
0.7923 
0.7921 
0.7919 
0.7918 

29 

28 
27 
26 
25 
24 
23 

21 

40 

0.6111 

0.7720 

1-2954 

0.7916 

20 

4i 
42 

43 
44 
45 
46 

47 
48 
49 

0.6113 
0.6115 
0.6118 
0.6120 
0.6122 
0.6124 
0:6127 
0.6129 
0.6131 

0.7724 
0.7729 
0-7734 
0.7738 
0.7743 
0-7747 
0-7752 
0-7757 
0.7761 

1.2946 
1.2938 
1.2931 
1-2923 
1.2915 
1.2907 

1.2900 
1.2892 
1.2884 

0.7914 
0.7912 
0.7910 
0.7909 
0.7907 
0.7905 
0.7903 
0.7902 
0.7900 

19 

18 

17 
16 
15 
14 
13 

12 
II 

50 

0.6134 

0.7766 

1.2870 

0.7898 

10 

51 

52 
53 
54 
55 
56 

57 
58 
59 

0.6136 
0.6138 
0.6141 
0.6143 
0.6145 
0.6147 
0.6150 
0.6152 
0.6154 

0.7771 

0-7775 
0.7780 

0-7785 
0.7789 
0-7794 

0.7799 
0.7803 
0.7808 

1.2869 
1.2861 
1-2853 
1.2846 
1.2838 
1.2830 
1.2822 
1.2815 
1.2807 

0.7896 
0.7894 
0.7893 
0.7891 
0.7889 
0.7887 
0.7885 
0.7884 
0.7882. 

9 

8 

7 
6 
.5 
4 
3 

2 

I 

60 

0.6157 

0.7813 

1.2799 

0.7880 

0 

Cos 

Cot 

Tan 

Sin 

' 

*143°  233°  *323° 


NATURAL 


>142°  232°  *322° 


*128°  218°  *308°   38' 


NATURAL 


39< 


127 

*129°  219°  *309° 


1 

Sin 

Tan 

Cot 

Cos 

0 
I 

2 

3 
4 

5 
6 

7 
8 

9 
10 
ii 

12 
13 
14 
15 

16 

17 

18 

*9 

20 

21 
22 

23 
24 
25 
26 

27 

28 

29 

30 

3i 

32 
33 
34 

35 
36 

37 
38 
39 
40 

4i 

42 

43 
44 
45 
46 

47 

48 

49 
50 

5i 

52 
53 
54 
55 
56 

57 
58 
59 
60 

0.6157 

0.7813 

1.2799 

0.7880 

60 

59 
58 
57 
56 
55 
54 

53 
52 
51 
50 

49 

48 

47 
46 
45 
44 

43 
42 
41 
40 

39 
38 
37 
36 
35 
34 

33 
32 
3i 
30 

29 

28 

27 
26 

25 
24 

23 

22 
21 

20 

19 

18 

17 
16 
15 
14 
13 

12 
II 
10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.6159 
0.6161 
0.6163 
0.6166 
0.6168 
0.6170 
0.6173 
0.6175 
0.6177 

0.7818 
0.7822 
0.7827 
0.7832 
0.7836 
0.7841 
0.7846 
0.7850 
0.7855 

1.2792 
1.2784 
1.2776 
1.2769 
1.2761 
1-2753 
1.2746 
1.2738 
1.2731 

0.7878 
0.7877 
0.7875 

0.7873 
0.7871 
0.7869 
0.7868 
0.7866 
0.7864 

0.6180 

0.7860 

1.2723 

0.7862 

0.6182 
0.6184 
0.6186 
0.6189 
0.6191 
0.6193 
0.6196 
0.6198 
0.6200 

0.7865 
0.7869 
0.7874 
0.7879 
0.7883 
0.7888 
0.7893 
0.7898 
0.7902 

1.2715 
1.2708 
1.2700 
1.2693 
1.2685 
1.2677 
1.2670 
1.2662 
1.2655 

0.7860 
0.7859 

0.7857 
0.7855 
0.7853 
0.7851 

0.7850 
0.7848 
0.7846 

0.6202 

0.7907 

1.2647 

0.7844 

0.6205 
0.6207 
0.6209 
0.6211 
0.6214 
0.6216 
0.6218 
0.6221 
0.6223 

0.7912 
0.7916 
0.7921 
0.7926 
0.7931 
0-7935 
0.7940 

0.794! 
0.7950 

1.2640 
1.2632 
1.2624 
1.2617 
1.2609 
1.2602 
1.2594 
1.2587 
1-2579 

0.7842 
0.7841 
0.7839 

0.7837 
0.7835 
0.7833 
0.7832 
0.7830 
0.7828 

0.6225 

0.7954 

1.2572 

0.7826 

0.6227 
0.6230 
0.6232 
0.6234 
0.6237 
0.6239 
0.6241 
0.6243 
0.6246 

0.7959 
0.7964 
0.7969 

0.7973 

0.7978 
0.7983 
0.7988 
0.7992 
0.7997 

1.2564 

1-2557 
1.2549 

1.2542 

1-2534 
1.2527 

1.2519 
1.2512 

1.2504 

0.7824 
0.7822 
0.7821 
0.7819 
0.7817 
0.7815 

0.7813 
0.7812 
0.7810 

0.6248 

0.8002 

1.2497 

0.7808 

0.6250 
0.6252 
0.6255 
0.6257 
0.6259 
0.6262 
0.6264 
0.6266 
0.6268 

0.8007 
0.8012 
0.8016 
0.8621 
0.8026 
0.8031 
0.8035 
0.8040 
0.8045 

1.2489 
1.2482 
1.2475 
1.2467 
1.2460 
1.2452 
1.2445 
1.2437 
1.2430 

0.7806 
0.7804 
0.7802 
0.7801 

0-7799 
0.7797 

o.7795 
0.7793 
0.7792 

0.6271 

0.8050 

1.2423 

0.7790 

0.6273 
0.6275 
0.6277 
0.6280 
0.6282 
0.6284 
0.6286 
0.6289 
0.6291 

0.8055 
0.8059 
0.8064 
0.8069 
0.8074 
0.8079 
0.8083 
0.8088 
0.8093 

1.2415 
1.2408 
1.2401 

1.2393 

1.2386 
1.2378 

1.2371 
1.2364 
1.2356 

0.7788 
0.7786 
0.7784 
0.7782 
0.7781 
0.7779 
0.7777 
0-7775 
•0.7773 

0.6293 

0:8098 

1.2349 

0.7771 

Cos 

Cot 

Tan 

Sin 

' 

/ 

Sin 

Tan 

Cot 

Cos 

0 

0.6293 

0.8098 

•2349 

0.7771 

60 

I 

2 

3 
4 
5 
6 

7 
8 

9 

0.6295 
0.6298 
0.6300 
0.6302 
0.6305 
0.6307 
0.6309 
0.6311 
0.6314 

0.8103 
0.8107 
0.8112 
0.8117 
0.8122 
0.8127 
0.8132 
0.8136 
0.8141 

.2342 
.2334 
.2327 

.2320 
.2312 
.2305 
.2298 
.2290 
.2283 

0.7770 
0.7768 
0.7766 
0.7764 
0.7762 
0.7760 

0-7759 
0.7757 
0.7755" 

59 
58 
57 
56 
55 
54 

53 
52 
5i 

10 

0.6316 

0.8146 

.2276 

o.7753 

50 

ii 

12 
13 
14 

15 

16 

17 

18 

J9 

0.6318 
0.6320 
0.6323 
0.6325 
0.6327 
0.6329 
0.6332 
0.6334 
0.6336 

0.8151 
0.8156 
0.8161 
0.8165 
0.8170 
0.8175 
0.8180 
0.8185 
0.8190 

.2268 
.2261 
.2254 
.2247 
.2239 
.2232 
.2225 
.2218 
.2210 

o.775i 
0.7749 
0.7748 
0.7746 
0.7744 
0.7742 
0.7740 
0.7738 
0-7737 

49 

48 

47 
46 
45 
44 

43 
42 
41 

20 

0.6338 

0.8195 

.2203 

0.7735 

40 

21 
22 
23 
24 
25 
2(3 

27 
28 
29 

0.6341 
0.6343 
0.6345 
0.6347 
0.6350 
0.6352 

0.6354 
0.6356 
0.6359 

0.8199 
0.8204 
0.8209 
0.8214 
0.8219 
0.8224 
0.8229 
0.8234 
0.8238 

.2196 
.2189 
.2181 
.2174 
.2167 
.2160 

•2153 
.2145 
.2138 

0-7733 
0.7731 
0.7729 

0.7727 
0.7725 
0.7724 
0.7722 
0.7720 
0.7718 

39 
38 
37 
36 
35 
34 
33 
32 
31 

30 

0.6361 

0.8243 

.2131 

0.7716 

30 

31 

32 

33 
34 

35 
36 

37 
38 
39 

0.6363 
0.6365 
0.6368 
0.6370 
0.6372 
0.6374 
0.6376 
0.6379 
0.6381 

0.8248 
0.8253 
0.8258 
0.8263 
0.8268 
0.8273 
0.8278 
0.8283 
0.8287 

.2124 
.2117 
.2109 

.2102 
.2095 
.2088 
.208l 
.2074 
.2O66 

0.7714 
0.7713 
0.7711 
0.7709 
0.7707 
0.7705 

0.7703 
0.7701 
0.7700 

29 
28 
27 
26 

25 
24 

23 

22 
21 

40 

0.6383 

0.8292 

.2059 

0.7698 

20 

4i 

42 

43 
44 
45 
46 

47 
48 

49 

0.6385 
0.6388 
0.6390 
0.6392 
0.6394 
0.6397 
0.6399 
0.6401 
0.6403 

0.8297 
0.8302 
0.8307 
0.8312 
0.8317 
0.8322 
0.8327 
0.8332 
0.8337 

.2O52 
.2045 
.2038 
.2031 
.2O24 
.2017 
.2OO9 
.2OO2 
•1995 

0.7696 
0.7694 
0.7692 
0.7690 
0.7688 
0.7687 
0.7685 
0.7683 
0.7681 

19 

18 

17 

16 
15 
14 
13 

12 
II 

50 

0.6406 

0.8342 

.1988 

0.7679 

10 

5i 

52 

53 
54 
55 
56 

57 
58 
59 

0.6408 
0.6410 
0.6412 

0.6414 
0.6417 
0.6419 
0.6421 
0.6423 
0.6426 

0.8346 
0.8351 
0.8356 
0.8361 
0.8366 
0.8371 
0.8376 
0.8381 
0.8386 

.1981 
.1974 
.1967 

.1960 

•1953 
.1946 

.1939 
.1932 

.1925 

0.7677 

0.7675 
0.7674 

0.7672 
0.7670 
0.7668 
0.7666 
0.7664 
0.7662 

I 

1 

6 
5 
4* 
3 

2 

I 

60 

0.6428 

0.8391 

I.igiS 

0.7660 

0 

Cos 

Cot 

Tan 

Sin 

' 

141°  231°  *321° 


NATURAL 


50' 


*140°  230°  *320° 


128 

*130°    220°  *310° 


40C 


NATURAL 


41°    *131°    221°  *311° 


' 

Sin 

Tan 

Cot 

Cos 

0 

0.6428 

0.8391 

1.1918 

0.7660 

60 

I 

2 

3 
4 

6 

7 
8 

9 

0.6430 
0.6432 
0.6435 
0.6437 
0.6439 
0.6441 
0.6443 
0.6446 
0.6448 

0.8396 
0.8401 
0.8406 
0.8411 
0.8416 
,  0.8421 
0.8426 
0.8431 
0.8436 

1.1910 

1.1903 
1.1896 

1.1889 
1.1882 
1.1875 
1.1868 
1.1861 
1.1854 

0.7659 
0.7657 
0.7655 

0.7653 
0.7651 
0.7649 

0.7647 
0.7645 
0.7644 

59 
58 
57 
56 
55 
54 
53 
52 
51 

10 

0.6450 

0.8441 

1.1847 

0.7642 

50 

ii 

12 

13 
14 
15 
16 

17 

18 

19 

0.6452 
0.6455 
0.6457 
0.6459 
0.6461 
0.6463 
0.6466 
0.6468 
0.6470 

0.8446 
0.8451 
0.8456 
0.8461 
0.8466 
0.8471 
0.8476 
0.8481 
0.8486 

1.1840 

1.1833 
1.1826 

1.1819 
1.  1812 
1.  1806 
1.1799 
1.1792 
1.1785 

0.7640 
0.7638 
0.7636 
0.7634 
0.7632 
0.7630 
0.7629 
0.7627 
0.7625 

49 

48 

47 
46 
45 
44 

43 
42 
4i 

20 

0.6472 

0.8491 

1.1778 

0.7623 

40 

21 
22 
23 
24 

25 
26 

27 

28 

29 

0.6475 
0.6477 
0.6479 
0.6481 
0.6483 
0.6486 
0.6488 
0.6490 
0.6492 

0.8496 
0.8501 
0.8506 
0.8511 
0.8516 
0.8521 
0.8526 
0.8531 
0.8536 

1.1771 
1.1764 
I-I757 
1.1750 
I.I743 
1.1736 
1.1729 
1.1722 
1.1715 

0.7621 
0.7619 
0.7617 
0.7615 
0.7613 
0.7612 
0.7610 
0.7608 
0.7606 

39 
38 
37 
36 
35 
34 

33 
32 
3i 

30 

0.6494 

0.8541 

1.1708 

0.7604 

80 

3i 
32 
33 
34 
35 
36 

37 
38 
39 

0.6497 
0.6499 
0.6501 
0.6503 
0.6506 
0.6508 
0.6510 
0.6512 
0.6514 

0.8546 
0.8551 
0.8556 
0.8561 
0.8566 
0.8571 
0.8576 
0.8581 
0.8586 

1.1702 
1.1695 

1.1688 

1.1681 
1.1674 
1.1667 
1.  1660 
1-1653 
1.1647 

0.7602 
0.7600 
0.7598 
0.7596 
0-7595 
0-7593 

0-7591 
0.7589 

0.7587 

29 
28 

27 
26 

25 
24 
23 

22 
21 

40 

0.6517 

0.8591 

1.1640 

0.7585 

20 

4i 
42 
43 
44 

45 
46 

47 
48 
49 

0.6519 
0.6521 
0.6523 
0.6525 
0.6528 
0.6530 
0.6532 
0.6534 
0.6536 

0.8596 
0.8601 
0.8606 
0.8611 
0.8617 
0.8622 
0.8627 
0.8632 
0.8637 

1.1633 
1.1626 
1.1619 
1.  1612 
1.  1606 
I-I599 
1.1592 

1-1585 
1.1578 

0.7583 
0.7581 

0-7579 
0.7578 
0.7576 
0-7574 
0.7572 
0-7570 
0.7568 

19 

18 
17 
16 
15 
14 
13 

12 
II 

50 

0.6539 

0.8642 

I-I57I 

0.7566 

10 

5i 
52 
53 
54 
55 
56 

57 
58 
59 

0.6541 
0.6543 
0.6545 
0.6547 
0.6550 
0.6552 

0.6554 
0.6556 
0.6558 

0.8647 
0.8652 
0.8657 
0.8662 
0.8667 
0.8672 
0.8678 
0.8683 
0.8688 

1-1565 
I.I558 
I.I55I 

I-I544 

1.1538 

I.I53I 
1.1524 
1.1517 
1.1510 

0.7564 
0.7562 
0.7560 

0-7559 
0-7557 
0-7555 
0-7553 
0.7551 
0.7549 

9 

8 

7 
6 
5 
4 
3 

2 
I 

60 

0.6561 

0.8693 

1.1504 

0-7547 

0 

Cos 

Cot 

Tan 

Sin 

' 

' 

Sin 

Tan 

Cot 

Cos 

0 

0.6561 

0.8693 

.1504 

0-7547 

60 

i 

2 

3 
4 

6 

8 
9 

0.6563 
0.6565 
0.6567 
0.6569 
0.6572 
0.6574 
0.6576 
0.6578 
0.6580 

0.8698 
0.8703 
0.8708 
0.8713 
0.8718 
0.8724 
0.8729 
0.8734 
0.8739 

.1497 
.1490 
.1483 

.1477 
.1470 
.1463 
.1456 
.1450 
.1443 

o.7545 
0-7543 
o.754i 

o.7539 
0.7538 
0.7536 

•  0.7534 
0.7532 
0.7530 

59 
58 
57 
56 
55 
54 

53 
52 
5i 

10 

0.6583 

0.8744 

.1436 

0.7528 

50 

ii 

12 
13 
14 
15 

16 

17 

18 

*9 

0.6585 
0.6587 
0.6589 
0.6591 
0.6593 
0.6596 
0.6598 
0.6600 
0.6602 

0.8749 

0.8754 
0.8759 

0.8765 
0.8770 
0.877! 
0.8780 
0.8785 
0.8790 

.1430 
.1423 
.1416 
.1410 
.1403 
.1396 
.1389 

.1383 
.1376 

0.7526 
0.7524 
0.7522 

0.7520 
0.7518 
0.7516 

0.7515 
0.7513 
0.7511 

49 

48 

47 
46 
45 
44 

43 
42 
41 

20 

0.6604 

0.8796 

.1369 

0.7509 

40 

21 
22 
23 
24 
25 
26 

27 
28 

29 

0.6607 
0.6609 
0.6611 
0.6613 
0.6615 
0.6617 
0.6620 
0.6622 
0.6624 

0.8801 
0.8806 
0.8811 
0.8816 
0.8821 
0.8827 
0.8832 
0.8837 
0.8842 

.1363 
•1356 
•1349 
•1343 
•1336 
.1329 

•1323 
.1316 
.1310 

0.7507 
0.7505 
0.7503 
0.7501 
0.7499 
0.7497 

0-7495 
0-7493 
0.7491 

39 
38 
37 
36 
35 
34 
33 
32 
3i 

30 

0.6626 

0.8847 

•  1303 

0.7490 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 

0.6628 
0.6631 
0.6633 
0.6635 
0.6637 
0.6639 
0.6641 
0.6644 
0.6646 

0.8852 
0.8858 
0.8863 
0.8868 
0.8873 
0.8878 
0.8884 
0.8889 
0.8894 

.1296 
.1290 
.1283 
.1276 
.1270 
.1263 
.1257 
.1250 
.1243 

0.7488 
0.7486 
0.7484 
0.7482 
0.7480 
0.7478 
0.7476 
0.7474 
0.7472 

29 

28 

27 
26 

25 
24 

23 

21 

40 

0.6648 

0.8899 

•1237 

0.7470 

20 

4i 
42 
43 
44 

45 
46 

47 
48 
49 

0.6650 
0.6652 
0.6654 
0.6657 
0.6659 
0.6661 
0.6663 
0.6665 
0.6667 

0.8904 
0.8910 
0.8915 
0.8920 
0.8925 
0.8931 
0.8936 
0.8941 
0.8946 

.1230 
.1224 
.1217 

.1211 
.I2O4 
.1197 
.Iigl 
.1184 
.1178 

0.7468 
0.7466 
0.7464 

0.7463 
0.7461 

0.7459 
0-7457 
0-7455 
0-7453 

«9 

18 

17 
16 
15 
14 
13 

12 
II 

50 

0.6670 

0.8952 

.1171 

o.745i 

10 

5i 
52 
53 
54 
55 
56 

57 
58 
59 

0.6672 
0.6674 
0.6676 
0.6678 
0.6680 
0.6683 
0.6685 
0.6687 
0.6689 

0.8957 
0.8962 
0.8967 
0.8972 
0.8978 
0.8983 
0.8988 
0.8994 
0.8999 

.1165 
.1158 
.1152 
•1145 
.1139 
.1132 

.1126 
.1119 
.1113 

0.7449 

0-7447 
0.7445 

0-7443 
0.7441 

0-7439 
0.7437 
0.7435 
0-7433 

9 

8 

7 
6 
5 
4 
3 

2 

I 

60 

0.6691 

0.9004 

.1  106 

0.7431 

0 

Cos 

Cot 

Tan 

Sin 

' 

*139°  229°  *319°   49' 


NATURAL 


*138°  228°  *318° 


*132°  222°  *312°       42' 


NATURAL 


43° 


129 
*133°  223°  *313° 


' 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 

6 

7 
8 

9 
10 
ii 

12 
13 
14 
15 

16 

17 

18 

J9 
20 

21 
22 
2_ 

24 

25 
26 

27 
28 
29 

30 

3i 

32 
33 
34 
35 
36 

37 
38 
39 
40 

4i 
42 
43 
44 
45 
46 

47 
48 

49 
50 

5i 

52 

53 
54 
55 
56 

57 
58 
59 
60 

0.6691 

0.9004 

1.1106 

0.7431 

60 

59 
58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 
47 
46 
45 
44 

43 

42 
41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
3i 
30 
29 
28 
27 
26 
25 
24 
23 

21 

20 

J9 
18 

17 
16 
15 
14 
13 

12 
II 
10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.6693 
0.6696 
0.6698 
0.6700 
0.6702 
0.6704 
0.6706 
0.6709 
0.6711 

0.9009 
0.9015 
0.9020 
0.9025 
0.9030 
0.9036 
0.9041 
0.9046 
0.9052 

I.IIOO 

1.1093 
1.1087 
1.1080 
1.1074 
1.1067 
1.1061 
1.1054 
1.1048 

0.7430 
0.7428 
0.7426 
0.7424 
0.7422 
0.7420 
0.7418 
0.7416 
0.7414 

0.6713 

0.9057 

1.1041 

0.7412 

0.6715 
0.6717 
0.6719 
0.6722 
0.6724 
0.6726 
0.6728 
0.6730 
0.6732 

0.9062 
0.9067 
0.9073 
0.9078 
0.9083 
0.9089 
0.9094 
0.9099 
0.9105 

1-1035 
1.1028 

1.  1022 

1.1016 

I.IOOg 
I.I003 
1.0996 
I.OggO 
1.0983 

0.7410 
.  0.7408 
0.7406 
0.7404 
0.7402 
0.7400 

0.7398 
0.7396 
0-7394 

0.6734 

0.9110 

1.0977 

0.7392 

0.5737 
0.6739 
0.6741 
0.6743 
0.6745 
0.6747 
0.6749 
0.6752 
0.6754 

0.9115 
0.9121 
0.9126 
0.9131 

0.9137 
0.9142 

0.9147 

0.9153 
0.9158 

I.097I 
1.0964 
1.0958 
I.095I 

1.0945 
1.0939 

1.0932 

1.0926 
1.0919 

0.7390 
0.7388 
0.7387 
0.7385 
0.7383 
0.7381 

o.7379 
0,7377 
0.7375 

0.6756 

0.9163 

1.0913 

0-7373 

0.6758 
0.6760 
0.6762 
0.6764 
0.6767 
0.6769 
0.6771 
0.6773 
0.6775 

0.9169 
0.9174 
0.9179 
0.9185 
0.9190 
0.9195 
0.9201 
0.9206 
0.9212 

1.0907 

1.0900 
1.0894 

1.0888 

i.oSSi 
1.0875 
1.0869 
1.0862 
1.0856 

0.7371 
0.7369 
0.7367 

0.7365 
0.7363 
0.7361 

0.7359 
0-7357 
0-7355 

0.6777 

0.9217 

1.0850 

0-7353 

0.6779 
0.6782 
0.6784 
0.6786 
0.6788 
0.6790 
0.6792 
0.6794 
0.6797 

0.9222 
0.9228 
0.9233 
0.9239 
0.9244 
0.9249 

0.9255 
0.9260 
0.9266 

1.0843 
1.0837 
1.0831 
1.0824 
1.0818 
1.0812 
1.0805 

1.0799 
1.0793 

0-7351 
0-7349 
0-7347 
0-7345 
0.7343 
o.734i 
0-7339 
0.7337 
0-7335 

0.6799 

0.9271 

1.0786 

0-7333 

0.6801 
0.6803 
0.6805 
0.6807 
0.6809 
o.68n 
0.6814 
0.68  1  6 
0.6818 

0.9276 
0.9282 
0.9287 
0.9293 
0.9298 
0.9303 
0.9309 
0.9314 
0.9320 

1.0780 

1.0774 

1.0768 
1.0761 

1.0755 
1.0749 

1.0742 
1.0736 
1.0730 

o.733i 
0.7329 
0.7327 

0.7325 
0.7323 
0.7321 

0.7319 

0.7318 
0.7316 

0.6820 

0.9325 

1.0724 

o.73i4 

Cos 

Cot 

Tan 

Sin 

' 

*137°  227°  *317° 


NATURAL 


' 

Sin 

Tan 

Cot 

Cos 

0 

I 

2 

3 
4 

6 

8 
9 
10 
ii 

12 
13 
14 
15 

16 

17 

18 

!9 

20 

22 
23 
24 
25 
26 

27 
28 
29 

30 

3i 
32 
33 
34 

35 
36 

37 
38 
39 
40 

4i 
42 

43 
44 
45 
46 

47 
48 

%49 
50 

5i 
52 
53 
54 
55 
56 

57 
58 
59 
60 

0.6820 

0.9325 

1.0724 

0.7314 

60 

59 
58 
57 
56 
55 
54 

53 
52 
5i 
50 

49 
48 

47 
46 
45 
44 

43 
42 

4i 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 

28 

27 
26 

25 
24 

23 

22 
21 
20 
19 

18 
17 
16 
15 
14 
13 

12 
II 

10 

9 

8 

7 
6 
5 
4 
3 

2 

I 

0 

0.6822 
0.6824 
0.6826 
0.6828 
0.6831 
0.6833 
0.6835 
0.6837 
0.6839 

0.9331 
0.9336 

0.9341 

o.9347 
0.9352 
0.9358 
0.9363 
0.9369 
0-9374 

1.0717 
1.0711 
1.0705 
1.0699 
1.0692 
1.0686 
i.  0680 
1.0674 
1.0668 

0.7312 
0.7310 
0.7308 
0.7306 
0.7304 
0.7302 
0.7300 
0.7298 
0.7296 

0.6841 

0.9380 

i.  066  1 

0.7294 

0.6843 
0.6845 
0.6848 
0.6850 
0.6852 
0.6854 
0.6856 
0.6858 
0.6860 

0.9385 
0.9391 
0^96 
0.9402 
0.9407 
0.9413 
0.9418 
0.9424 
0.9429 

1.0655 
1.0649 
1.0643 
1.0637 
1.0630 
1.0624 
i.  0618 
1.0612 

1.  0606 

0.7292 
0.7290 
0.7288 
0.7286 
0.7284 
0.7282 
0.7280 
0.7278 
0.7276 

0.6862 

0.9435 

1.0599 

0.7274 

0.6865 
0.6867 
0.6869 
0.6871 
0.6873 
0.6875 
0.6877 
0.6879 
0.6881 

0.9440 
0.9446 
0.9451 

o.9457 
0.9462 
0.9468 

0-9473 
0.9479 
0.9484 

1.0593 
1.0587 

1.0581 

1.0575 

1.0569 
1.0562 

1.0556 
1.0550 
1.0544 

0.7272 
0.7270 
0.7268 
0.7266 
0.7264 
0.7262 
0.7260 
0.7258 
0.7256 

0.6884 

0.9490 

1.0538 

0.7254 

0.6886 
0.6888 
0.6890 
0.6892 
0.6894 
0.6896 
0.6898 
0.6900 
0.6903 

0.9495 
0.9501 
0.9506 
0.9512 

0.9517 
0.9523 

0.9528 

0-9534 
0.9540 

1.0532 

1.0526 

1.0519 

1.0513 
1.0507 

1.0501 

1.0495 
1.0489 
1.0483 

0.7252 
0.7250 
0.7248 
0.7246 
0.7244 
0.7242 
0.7240 
0.7238 
0.7236 

0.6905 

0-9545 

1.0477 

0.7234 

0.6907 
0.6909 
o.6qn 
0.6913 
0.6915 
0.6917 
0.6919 
0.6921 
0.6924 

0.9551 
0.9556 
0.9562 
0.9567 

0.9573 
0.9578 

0.9584 
0.9590 
0.9595 

1.0470 

1.0464 
1.0458 

1.0452 
1.0446 

1.0440 

1.0434 

1.0428 
1.0422 

0.7232 
0.7230 
0.7228 
0.7226 
0.7224 
0.7222 
0.7220 
0.7218 
0.7216 

0.6926 

0.9601 

1.0416 

0.7214 

0.6928 
0.6930 
0.6932 
0.6934 
0.6936 
0.6938 
0.6940 
0.6942 
0.6944 

0.9606 
0.9612 
0.9618 
0.9623 
0.9629 
0.9634 
0.9640 
0.9646 
0.9651 

1.0410 

1.0404 

1.0398 

1.0392 
1.0385 

1.0379 
1.0373 

1.0367 
1.0361 

0.7212 
0.7210 
0.7208 
0.7206 
0.7203 
0.7201 

0.7199 
0.7197 

0.7195 

0.6947 

0.9657 

1.0355 

0.7193 

Cos   Cot 

Tan 

Sin 

' 

*136°  226°  *316° 


130 


NATURAL        44°    *134°  224°  *314° 


' 

Sin 

Tan 

Cot 

Cos 

0 

0.6947 

0.9657 

1-0355 

0.7193 

60 

I 

2 

3 
4 

6 

7 
8 

9 

0.6949 
0.6951 
0.6953 
0.6955 
0.6957 
0.6959 

0.6961 
0.6963 
0.6965 

0.9663 
0.9668 
0.9674 
0.9679 
0.9685 
0.9691 
0.9696 
0.9702 
0.9708 

1.0349 

1.0343 
1.0337 

1.0331 
1.0325 
1.0319 

1.0313 
1.0307 
1.0301 

0.7191 
0.7189 
0.7187 
0.7185 
0.7183 
0.7181 
0.7179 
0.7177 
0.7175 

59 

58 
57 
56 
55 
54 

53 
52 
Si 

10 

0.6967 

0.9713- 

1.0295 

0.7173 

50 

ii 

12 
13 
14 
15 

16 

17 
18 

19 

0.6970 
0.6972 
0.6974 
0.6976 
0.6978 
0.6980 
0.6982 
0.6984 
0.6986 

0.9719 
0.9725 
0.9730 

0.9736 
0.9742 
0.9747 

0.9753 
0.9759 
0.9764 

1.0289 
1.0283 
1.0277 
1.0271 
1.0265 
1.0259 
1.0253 
1.0247 
1.0241 

0.7171 
0.7169 
0.7167 
0.7165 
0.7163 
0.7161 

0.7159 
0.7157 
0.7155 

49 

48 

47 
46 
45 
44 

43 
42 

41 

20 

0.6988 

0.9770 

1.0235 

0.7153 

40 

21 

22 
23 
24 

25 
26 

27 

28 

29 

0.6990 
0.6992 
0.6995 
0.6997 
06999 
0.7001 
0.7003 
0.7005 
0.7007 

0.9776 
0.9781 
0.9787 

0.9793 
0.9798 
0.9804 
0.9810 
0.9816 
0.9821 

1.0230 
1.0224 
1.0218 

I.02I2 
1.0206 
I.O2OO 
I.OI94 
I.OI88 
I.OI82 

0.7151 
0.7149 
0.7147 

0.714! 
0.7143 
0.7141 
0.7139 
0.7137 
0.713! 

39 
38 
37 
36 
35 
34 
33 
32 
31 

30 

0.7009 

0.9827 

I.OI76 

0.7133 

30 

3i 
32 
33 
34 
35 
36 

37 
38 
39 

0.7011 
0.7013 
0.7015 
0.7017 
0.7019 
0.7022 
0.7024 
0.7026 
0.7028 

0.9833 
0.9838 
0.9844 
0.9850 
0.9856 
0.9861 
0.9867 
0.9873 
0.9879 

I.OI70 
I.OI04 
I.OI58 
I.OI52 
I.OI47 
I.OI4I 

I-OI35 
I.OI29 
I.OI23 

0.7130 
0.7128 
0.7126 
0.7124 
0,7122 
0.7120 
0.7118 
0.7116 
0.7114 

29 

28 

27 
26 
25 
24 
23 

22 

21 

40 

0.7030 

0.9884 

I.OII7 

0.7112 

20 

4i 
42 

43 

44 
45 
46 

47 
48 
49 

0.7032 
0.7034 
0.7036 
0.7038 
0.7040 
0.7042 
0.7044 
o.  7046 
0.7048 

0.9890 
0.9896 
0.9902 
0.9907 
0.9913 
0.9919 
0.9925 
0.9930 
0.9936 

I.OIII 
.0105 
.0099 
.0094 
.0088 
.0082 
.0076 
.OO7O 
I.OO64 

0.7110 
0.7108 
0.7106 
0.7104 
0.7102 
0.7100 
0.7098 
0.7096 
0.7094 

19 
18 
17 
16 
15 
14 
13 

12 
II 

50 

0.7050 

0.9942 

1.0058 

0.7092 

10 

5i 
52 

53 
54 
55 
56 

57 
58 
59 

0.7053 
0.7055 
0.7057 
0.7059 
0.7061 
0.7063 
0.7065 
0.7067 
0.7069 

0.9948 
0.9954 
0.9959 
0.9965 
0.9971 
0.9977 
0.9983 
0.9988 
0.9994 

1.0052 
1.0047 
I.OO4I 
1.0035 
I.OO29 
1.0023 
I.OOI7 
I.OOI2 

i.coo6 

0.7090 
0.7088 
0.7085 
0.7083 
0.7081 
0.7079 
0.7077 
0.7075 
0.7073 

9 

8 

7 
6 
5 
4 
3 

2 

I 

60 

0.7071 

I.OOOO 

I.OOOO 

0.7071 

0 

Cos 

Cot 

Tan  j 

Sin 

' 

*135°  225°  *315°      45°    NATURAL 


131 


VI 

TABLE  OF  SQUARES, 

CUBES,  SQUARE  ROOTS  AND  CUBE  ROOTS 

OF 

WHOLE  NUMBERS  FROM  1  TO  1020. 

The  numbers  are  given  in    the  columns  headed  N,   their  squares, 

cubes,  square  roots  and  cube  roots  respectively  in  the  columns  headed  N2, 

N3.  -/N  and  f  N 

0—60 

X 

N2 

X3 

l/N 

ifN 

X 

X2 

X3 

i/N 

T^N 

0 

0 

0 

o.oooo 

0.0000 

30 

900 

27000 

5.4772 

3-1072 

i 

i 

i 

I.OOOO 

.0000 

31 

961 

29791 

5-5678 

3.I4I4 

2 

4 

8 

1.4142 

•2599 

32 

1024 

32768 

5-6569 

3.1748 

3 

9 

27 

1.7321 

.4422 

33 

1089 

35937 

5.7446 

3-2075 

4 

16 

64 

2.OOOO 

.5874 

34 

1156 

39304 

5.8310 

3.2396 

5 

25 

125 

2.2361 

.7100 

35 

1225 

42875 

5-9161 

3.2711 

6 

36 

216 

2.4495 

.8171 

36 

1296 

46656 

6.0000 

3-3019 

7 

49 

343 

2.6458 

1.9129 

37 

1369 

50653 

6.0828 

3.3322 

8 

64 

512 

2.8284 

2.0000 

38 

1444 

54872 

6.1644 

3.3620 

9 

81 

729 

3.0000 

2.0801 

39 

1521 

59319 

6.2450 

3.3912 

10 

IOO 

IOOO 

3.1623 

2.1544 

40 

1600 

64000 

6.3246 

3.4200 

ii 

121 

1331 

3.3166 

2.2240 

41 

1681 

68921 

6.4031 

34482 

12 

144 

1728 

3.4641 

2.2894 

42 

1764 

74088 

6.4807 

34760 

13 

169 

2197 

3.6056 

2.3513 

43 

1849 

79507 

6-5574 

3.5034 

14 

196 

2744 

3.7417 

2.4101 

44 

1936 

85184 

6.6332 

3.5303 

15 

225 

3375 

3.8730 

2.4662 

45 

2025 

91125 

6.7082 

3.5569 

16 

256 

4096 

4.0000 

2.5198 

46 

2116 

97336 

6.7823 

3-5830 

17 

289 

4913 

4.1231 

2.5713 

47 

2209 

103823 

6.8557 

3.6088 

18 

324 

5832 

4.2426 

2.6207 

48 

2304 

110592 

6.9282 

3.6342 

19 

361 

6859 

4.3589 

2.6684 

49 

2401 

117649 

7.0000 

3.6593 

20 

4OO 

8000 

4.4721 

2.7144 

50 

2500 

125000 

7.0711 

3.6840 

21 

441 

9261 

4.5826 

2.7589 

51 

2601 

132651 

7.1414 

3.7084 

22 

484 

10648 

4.6904 

2.8020 

52 

2704 

140608 

7.2111 

23 

529 

12167 

4.7958 

2.8439 

53 

2809 

148877 

7.2801 

3.7563 

24 

576 

13824 

4.8990 

2.8845 

54 

2916 

157464 

7.348! 

3.7798 

25 

625 

15625 

5.0000 

2.9240 

55 

3025 

166375 

7.4162 

3.8030 

26 

676 

17576 

5.0990 

2.9625 

56 

3136 

175616 

74833 

3-8259 

27 

729 

19683 

5.1962 

3.0000 

57 

3249 

185193 

7.5498 

3-8485 

28 

784 

219^2 

5.2915 

3.0366 

58 

3364 

195112 

7-6158 

3.8709 

29 

841 

24389 

5.3852 

3.0723 

59 

348i 

205379 

7.6811 

3.8930 

30 

9OO 

27000 

5.4772 

3.1072 

60 

3600 

216000 

7./460 

3.9149 

X 

X' 

X3 

75 

fN 

X 

X2 

X3 

l/N 

fN 

132 


60—180 


N 

N2 

N8 

1/N 

jni 

N 

N2 

N3 

1/N 

fN 

60 

61 
62 
63 
64 
65 
66 

67 
68 
69 
70 

7i 

72 
73 
74 
75 
76 

77 
78 
79 
80 
81 
82 
83 
84 
85 
86 

87 
88 
89 
90 

9i 
92 

93 
94 
95 
96 

97 
98 
99 
100 

IOI 
102 
103 
104 
105 
I  O6 

107 

108 
109 
110 
in 

112 
H3 
114 
H5 

116 
117 
118 
119 
120 

3600 

216000 

7.746o 

3.9149 

120 

121 
122 
123 
124 

125 
126 

127 
128 
I29 

130 

131 
132 

133 
134 
135 
136 

137 
138 
139 
140 
141 
142 
143 
144 

145 
146 

147 
148 
149 
150 

151 
152 
i53 
154 

i55 
156 

157 
158 
159 
160 
161 
162 
163 
164 

165 
1  66 

167 
1  68 
169 
170 
171 
172 
173 
i74 
i75 
176 

177 
178 
179 
180 

14400 

1728000 

10.9545 

4.9324 

3721 
3844 

3969 
4096 
4225 
4356 

4489 
4624 
4761 

226981 
238328 
250047 
262144 
274625 
287496 
300763 
314432 
328509 

7.8102 
7.8740 
7-9373 
8.0000 
8.0623 
8.1240 
8.1854 
8.2462 
8.3066 

3-9365 
3.9579 
3.9791 
4.0000 
4.0207 
4.0412 
4.0615 
4.0817 
4.1016 

14641 
14884 
15129 

15376 
15625 
15876 
16129 
16384 
16641 

1771561 
1815848 
1860867 
1906624 

1953125 
2000376 

2048383 
2097152 
2146689 

11.0000 

11.0454 
11.0905 

11.1355 

11.1803 
11.2250 
11.2694 

11.3137 
11.3578 

4.9461 
4-9597 
4-9732 
4.9866 
5.0000 
5-0133 
5.0265 

5-0397 
5.0528 

4900 

343000 

8.3666 

4.1213 

16900 

2197000 

11.4018 

5.0658 

5041 
5184 
5329 
5476 
5625 
5776 

5929 
6084 
6241 

357911 
373248 
389017 

405224 

421875 
438976 

456533 
474552 
493039 

8.4261 
84853 
8.5440 
8.6023 
8.6603 
8.7178 

8.7750 
8.8318 
8.8882 

4.1408 
4.1602 
4.1793 
4.1983 
4.2172 
4.2358 

4.2543 
4.2727 
4.2908 

17161 
17424 
17689 

17956 
18225 
18496 
18769 
19044 
19321 

2248091 
2299968 
2352637 
2406104 
2460375 
2515456 

2571353 
2628072 
2685619 

n-4455 
11.4891 
11.5326 

II.5758 
11.6190 
11.6619 
11.7047 
11.7473 
11.7898 

5.0788 
5.0916 
5-1045 

5.1172 
5.1299 
5.1426 

5.I55I 
5-1676 
5.1801 

6400 

512000 

8.9443 

4.3089 

19600 

2744000 

11.8322 

5-1925 

6561 
6724 
6889 
7056 
7225 
7396 
7569 
7744 
7921 

53I44I 
551368 
571787 
592704 
614125 
636056 

658503 
681472 
704969 

9.0000 

9-0554 
9.1104 

9.1652 
9.2195 
9-2736 

9.3274 
9.3808 
9.4340 

4.3267 
4.3445 
4.3621 

4-3795 
4.3968 
4.4140 
4.4310 
4.4480 
4.4647 

19881 
20164 
20449 
20736 
21025 
21316 
21609 
21904 

22201 

2803221 
2863288 
2924207 
2985984 
3048625 
3112136 

3176523 
3241792 
3307949 

11.8743 
11.9164 
11-9583 

I2.OOOO 
12.0416 
12.0830 
12.1244 
12.1655 
12.2066 

5.2048 
5.2171 

5-2293 

5-2415 
5.2536 
5-2656 

5.2776 
5.2896 
5-3015 

8100 

729000 

9.4868 

4.4814 

22500;     3375OOO 

12.2474 

5-3I33 

8281 
8464 
8649 
8836 
9025 
9216 
9409 
9604 
9801 

753571 
778688 
804357 

830584 
857375 
884736 

912673 
941192 
970299 

9-5394 
9.59I7 
9.6437 

9.6954 
9.7468 
9.7980 
9.8489 
9.8995 
9.9499 

4-4979 
4.5144 

4.5307 
4.5468 
4.5629 
4-5789 

4.5947 
4.6104 
4.6261 

228OI 
23104 
23409 

23716 
24025 
24336 
24649 
24964 
25281 

3442951 
3511808 

3581577 
3652264 

3723875 
3796416 

3869893 
3944312 
4019679 

12.2882 
12.3288 
12.3693 
12.4097 
12.4499 
12.4900 
12.5300 
12.5698 
12.6095 

5-3251 
5-3368 
5-3485 
5-36oi 
5.3717 
5-3832 

5.3947 
54061 
54175 

1  0000 

I  000000 

10.0000 

4.6416 

25600 

4096000 

12.6491 

54288 

I020I 
IO4O4 
10609 
I08l6 
II025 
II236 
II449 
II664 
Il88l 

1030301 
1061208 

1092727 

1124864 

1157625 

1191016 

1225043 
1259712 

1295029 

10.0499 
10.0995 
10.1489 
10.1980 
10.2470 
10.2956 
10.3441 
10.3923 
10.4403 

4.6570 

4-6723 
4.6875 

4.7027 
4.7177 
4.7326 

4-7475 
4.7622 

4.7769 

25921 
26244 
26569 
26896 
27225 
27556 
27889 
28224 
28561 

4173281 
4251528 
4330747 
4410944 
4492125 
4574296 

4657463 
4741632 
4826809 

12.6886 
12.7279 
12.7671 
12.8062 
12.8452 
12.8841 
12.9228 
12.9615 
13.0000 

5.4401 
5.4514 
5.4626 

5-4737 
54848 
5-4959 
5-5069 

5.5178 
5-5288 

I2IOO 

1331000 

10.4881 

4.7914 

28900 

4913000 

13.0384 

5-5397 

I232I 

12544 
12769 

12996 
13225 
13456 
13689 
13924 
I4l6l 

1367631 

1404928 

1442897 
1481544 
1520875 

1560896 
1601613 
1643032 
1685159 

10.5357 
10.5830 
10.6301 
10.6771 
10.7238 
10.7703 
10.8167 
10.8628 
10.9087 

4.8059 
4.8203 
4-8346 
4.8488 
4.8629 
4.8770 
4.8910 
4.9049 
4.9187 

29241 
29584 
29929 
3,0276 
30625 
30976 

31329 
31684 
32041 

5000211 
5088448 
5i777i7 
5268024 
5359375 
5451776 

5545233 
5639752 
5735339 

13-0767 
13.1149 
13-1529 

13.1909 
13.2288 
13.2665 

13.3041 
13.3417 
I3.379I 

5-5505 
5-5613 
5.5721 
5-5828 

5-5934 
5.6041 

5.6147 
5-6252 

5.6357 

I44OO 

10.9545 

4-9324 

32400 

5832000 

13.4164 

5.6462 

5 

N2 

tf3    |  v/N 

£H 

N 

N2 

Xs    |   i/N  |   fff 

133 


180—300 


N 

Na 

N3 

l/N 

fN 

N 

ET* 

N3 

1/N 

fN 

180 
181 
182 
183 

1,84 
185 
186 

187 
188 
189 
190 
191 
192 
193 
194 

195 
196 

197 
198 
199 
200 

201 
2O2 
2O3 
204 
205 
2O6 
207 
208 
2O9 

210 

211 
212 
213 
214 

215 
216 

217 

218 

219 

220 

221 
222 
223 
224 
225 
226 
227 
228 
229 

230 
231 
232 

233 
234 
235 
236 

237 
238 

239 
240 

32400 

5832000 

13.4164 

5.6462 

240 
241 
242 
243 
244 

245 
246 

247 
248 
249 
250 

251 
252 

253 
254 
255 
256 

257 

258 

259 
260 
261 
262 
263 
264 
265 
266 
267 
268 
269 
270 
271 
272 
273 
274 

275 
276 

277 
278 
279 
280 
281 
282 
283 
284 
285 
286 

287 
288 
289 
290 
291 
292 
293 
294 
295 
296 

297 

298 
299 
300 

57600 

13824000 

15.4919 

6.2145 

32761 
33124 
33489 
33856 
34225 
34596 

34969 
35344 
35721 

5929741 
6028568 
6128487 
6229504 
6331625 
6434856 

6539203 
6644672 
6751269 

134536 
13.4907 
13.5277 

13.5647 
13.6015 
13.6382 

I3-6748 
13.7113 
13.7477 

5.6567 
5.6671 
5.6774 

5-6877 
5.6980 
5.7083 

5.718! 
5.7287 
5.7388 

58081 
58564 
59°49 
59536 
60025 
60516 
61009 
61504 
62001 

13997521 
14172488 
14348907 
14526784 
14706125 
14886936 
15069223 
15252992 
15438249 

15-5242 
15.5563 
15.5885 
15.6205 
15-6525 
15.6844 
15-7162 
I5.7480 
15-7797 

6.2231 
6.2317 
6.2403 
6.2488 

6.2573 
6.2658 

6.2743 
6.2828 
6.2912 

36100 

6859000 

13.7840 

5.7489 

62500 

15625000 

15.8114 

6.2996 

36481 
36864 
37249 
37636 
38025 
38416 
38809 
39204 
39601 

6967871 
7077888 
7189057 

7301384 
7414875 
7529536 

7645373 
7762392 
7880599 

13.8203 
13-8564 
13.8924 

13-9284 
13.9642 
14.0000 

14.0357 
14.0712 
14.1067 

5.7590 
5.7690 
5.7790 
5.7890 

5.7989 
5.8088 

5.8186 
5-8285 
5.8383 

63001 
63504 
64009 

64516 
65025 
65536 
66049 
66564 
67081 

15813251 
16003008 
16194277 
16387064 

16581375 
16777216 

16974593 
17173512 
17373979 

15.8430 
15.8745 
15.9060 

15-9374 
15.9687 
16.0000 
16.0312 
16.0624 
16.0935 

6.3080 
6.3164 
6.3247 
6.3330 

6.3413 
6.3496 

6.3579 
6.3661 

6.3743 

40000 

8000000 

14.1421 

5.8480 

67600 

17576000 

16.1245 

6.3825 

40401 
40804 
41209 
41616 
42025 
42436 
42849 
43264 
43681 

8120601 
8242408 
8365427 
8489664 
8615125 
8741816 
8869743 
8998912 
9129329 

14.1774 
14.2127 
14.2478 
14.2829 
14.3178 
I4.3527 

14.3875 
14.4222 
14.4568 

5.8578 
5.8675 
5.8771 
5.8868 
5.8964 
5.9059 
5.9X55 
5-9250 
5-9345 

68121 
68644 
69169 
69696 
70225 
70756 
71289 
71824 
72361 

17779581 
17984728 
18191447 
18399744 
18609625 
18821096 

19034163 
19248832 
19465109 

16.155! 
16.1864 
16.2173 
16.2481 
16.2788 
16.3095 
16.3401 
16.3707 
16.4012 

6.3907 
6.3988 
6.4070 
6.4151 
6.4232 
6.4312 

6-4393 
6.4473 

6-4553 

44100 

9261000 

14.4914 

5-9439 

72900 

19683000 

16.4317 

6-4^33 

44521 
44944 
45309 

45796 
46225 
46656 
47089 

47524 
47961 

9393931 
9528128 

9663597 
9800344 

9938375 
10077696 

10218313 
10360232 
10503459 

14.5258 
14.5602 
14-5945 
14.6287 
14.6629 
14.6969 
14.7309 
14.7648 
14.7986 

5-9533 
5.9627 

5-9721 
5.9814 

5.9907 
6.0000 

6.0092 
6.0185 
6.0277 

7344i 
73984 
74529 
75076 
75625 
76176 
76729 
77284 
77841 

19902511 
20123648 
20346417 
20570824 
20796875 
21024576 

21253933 
21484952 
21717639 

16.4621 
16.4924 
16.5227 
16.5529 
16.5831 
16.6132 
16.6433 
16.6733 
16.7033 

6.4713 
6.4792 
6.4872 
6.4951 
6.5030 
6.5108 

6.5187 
6.5265 
6-5343 

48400 

10648000 

14.8324 

6.0368 

78400 

21952000 

16.7332 

6.5421 

48841 
49284 
49729 
50176 
50625 
51076 

51529 
51984 
52441 

10793861 
10941048 
11089567 
11239424 
11390625 
11543176 
11697083 
11852352 
12008989 

14.8661 
14.8997 
14.9332 
14.9666 
15.0000 
15.0333 
15.0665 
15.0997 
15-1327 

6.0459 
6.0550 
6.0641 
6.0732 
6.0822 
6.0912 

6.IOO2 

6.1091 
6.1180 

78961 

79524 
80089 

80656 
81225 
81796 
82369 
82944 
83521 

22188041 
22425768 
22665187 
22906304 
23149125 
23393656 

23639903 

23887872 
24137569 

16.7631 
16.7929 
16.8226 
16.8523 
16.8819 
16.9115 
16.9411 
16.9706 
17.0000 

6.5499 
6-5577 
6.5654 

6.5731 
6.5808 
6.5885 
6.5962 
6.6039 
6.6115 

52900 

12167000 

15.1658 

6.1269 

84100 

24389000 

17.0294 

6.6191 

5336i 
53824 
54289 

54756 
55225 
55696 
56169 
56644 
57121 

12326391 
12487168 
12649337 
12812904 
12977875 
13144256 
13312053 
13481272 
13651919 

15.1987 

15-2315 
15-2643 

15.2971 

15.3297 
15.3623 

15.3948 
15.4272 
154596 

6.1358 
6.1446 
6.1534 
§.1622 
6.1710 
6.1797 
6.1885 
6.1972 
6.2058 

84681 
85264 
85849 

86436 
87025 
87616 
88209 
88804 
89401 

24642171 
24897088 
25153757 
25412184 
25672375 
25934336 
26198073 
26463592 
26730899 

17.0587 
17.0880 
17.1172 
17.1464 
17.1756 
17.2047 

17.2337 
17.2627 
17.2916 

6.6267 
6.6343 
6.6419 
6.6494 
6.6569 
6.6644 
6.6719 
6.6794 
6.6869 

576oo 

13824000 

15.4919 

6.2145 

90000 

2  7OOOOOO 

17-3205 

6.6943 

K 

N2  |   N3 

V/N  |  f  N 

N 

X3 

Jfl 

T/N 

insr 

134 


300—420 


K 

w 

N8 

V/N 

rs 

N 

N2 

Ns 

'  T/N 

fN 

300 

301 
302 
303 
304 
305 
306 

307 
308 

309 
310 

3ii 
312 
3i3 
3i4 
3i5 
316 

3i7 
318 
319 
320 
321 
322 
323 
324 
325 
326 

327 

328 

329 
330 

33i 
332 
333 
334 
335 
336 

337 
338 
339 
340 

34i 
342 
343 
344 
345 
346 

347 
348 
349 
350 

35i 
352 
353 
354 
355 
356 

357 
358 
359 
360 

90000 

27000000 

17-3205 

6.6943 

360 

361 
362 
363 
364 
365 
366 

367 
368 

369 
370 

37i 
372 
373 
374 
375 
376 

377 
378 
379 
380 

38i 
382 
383 
384 
385 
386 

387 

388 

389 
390 

39i 
392 
393 
394 
395 
396 
397 
398 
399 
400 
401 
402 
403 
404 
405 
406 

407 
408 
409 
410 
411 
412 
413 
414 

415 
416 

417 
418 
419 
420 

129600 

46656000 

18.9737 

7.1138 

90601 
91204 

91809 

92416 

93025 
93636 

94249 

94864 

95481 

27270901 
27543608 
27818127 
28094464 
28372625 
28652616 
28934443 
29218112 
29503629 

17-3494 
17.3781 
17.4069 

174356 
17.4642 
17.4929 
17.5214 
17-5499 
17.5784 

6.7018 
6.7092 
6.7166 

6.7240 

6.7313 
6.7387 

6.7460 

6.7533 
6.7606 

130321 
131044 
131769 
132496 
133225 
133956 
134689 

135424 
136161 

47045881 
47437928 
47832147 
48228544 
48627125 
49027896 
49430863 
49836032 
50243409 

19.0000 
19.0263 
19.0526 
19.0788 
19.1050 
I9.I3H 
19.1572 

19.1833 
19.2094 

7.1204 
7.1269 
7.1335 
7.1400 
7.1466 
7.I53I 

7.1596 
7.1661 
7.1726 

96100 

29791000 

17.6068 

6.7679 

136900 

50653000 

I9.2354 

7.1791 

96721 

97344 
97969 

98596 
99225 
99856 
100489 
101124 
101761 

30080231 
30371328 
30664297 

30959144 
31255875 
31554496 
31855013 
32157432 
32461759 

17.6352 
17-6635 
17.6918 
17.7200 
17.7482 
17.7764 
17.8045 
17.8326 
17.8606 

6.7752 
6.7824 
6.7897 
6.7969 
6.8041 
6.8113 
6.8185 
6.8256 
6.8328 

137641 
138384 
139*29 
139876 
140625 
141376 
142129 
142884 
143641 

51064811 
51478848 
51895117 
52313624 
52734375 
53157376 
53582633 
54010152 
54439939 

19.2614 
19.2873 
19.3132 

I9.339I 
19.3649 
19.3907 
19.4165 
19.4422 
19.4679 

7.1855 
7.1920 

7.1984 
7.2048 
7.2112 
7-2177 
7.2240 
7.2304 
7.2368 

102400 

32768000 

17.8885 

6.8399 

144400 

54872000 

19.4936 

7.2432 

103041 
103684 
104329 
104976 
105625 
106276 
106929 
107584 
108241 

33076161 
33386248 
33698267 
34012224 
34328125 
34645976 
34965783 
35287552 
35611289 

17,9165 
17.9444 
17.9722 
18.0000 
18.0278 
18.0555 
18.0831 
18.1108 
18.1384 

6.8470 
,6.8541 
6.8612 
6.8683 

6.8753 
6.8824 

6.8894 
6.8964 
6.9034 

145161 
145924 
146689 

147456 
148225 
148996 
149769 
150544 
151321 

55306341 
55742968 
56181887 
56623104 
57066625 
57512456 
57960603 
58411072 
58863869 

19.5192 
19.5448 
19.5704 

19.5959 
19.6214 
19.6469 
19.6723 
19.6977 
19.7231 

7-2495 
7.2558 
7.2622 

7-2685 
7.2748 
7.2811 

7.2874 
7.2936 
7.2999 

108900 

35937000 

18.1659 

6.9104 

152100 

59319000 

19.7484 

7.3061 

109561 
110224 
110889 

111556 
112225 
112896 

113569 
114244 
114921 

36264691 
36594368 
36926037 
37259704 
37595375 
37933056 

38272753 
38614472 
38958219 

18.1934 
18.2209 
18.2483 
18.2757 
18.3030 
18.3303 

18.3576 
18.3848 
18.4120 

6.9174 
6.9244 
6.9313 
6.9382 
6.9451 
6.9521 
6.9589 
6.9658 
6.9727 

152881 
153664 
154449 
155236 
156025 
156816 
157609 
158404 
159201 

59776471 
60236288 
60698457 
61162984 
61629875 
62099136 
62570773 
63044792 
63521199 

19-7737 
19.7990 
19.8242 
19.8494 
19.8746 
19.8997 
19.9249 

19.9499 
19.9750 

7.3124 
7.3186 
7.3248 

7.3310 
7.3372 
7.3434 
7.3496 
7.3558 
7.3619 

115600 

39304000 

18.4391 

6.9795 

160000 

64000000 

20.0000 

7o68l 

116281 
116964 
117649 
118336 
119025 
119716 
120409 
121104 
121801 

39651821 
40001688 
40353607 
40707584 
41063625 
41421736 

41781923 
42144192 
42508549 

18.4662 
18.4932 
18.5203 

18.5472 
18.5742 
18.6011 
18.6279 
18.6548 
18.6815 

6.9864 
6.9932 
7.0000 

7.0068 
7.0136 
7.0203 
7.0271 
7.0338 
7.0406 

160801 
161604 
162409 
163216 
164025 
164836 
165649 
166464 
167281 

64481201 
64964808 
65450827 
65939264 
66430125 
66923416 
67419143 
67917312 
68417929 

2O.O25O 
20.0499 
2O.O749 
20.0998 
20.1246 
20.1494 
20.1742 
2O.I99O 
20.2237 

7.3742 
7o8o3 
7.3864 

7.3925 
7.3986 
74047 
7.4108 
7.4169 
7.4229 

122500 

42875000 

18.7083 

7.0473 

168100 

68921000 

20.2485 

74290 

123201 
123904 
124609 
125316 
126025 
126736 
127449 
128164 
128881 

43243551 
43614208 
43986977 
44361864 

44738875 
45118016 

45499293 
45882712 
46268279 

18.7350 
18.7617 
18.7883 
18.8149 
18.8414 
18.8680 
18.8944 
18.9209 
18.9473 

7.0540 
7.0607 
7.0674 
7.0740 
7.0807 
7.0873 
7.0940 
7.1006 
7.1072 

168921 
169744 
170569 

171396 
172225 
173056 
173889 
174724 
175561 

69426531 
69934528 
70444997 
70957944 
71473375 
71991296 

72511713 
73034632 
73560059 

20.2731 
20.2978 
20.3224 
20.3470 
20.3715 
20.3961 
20.4206 
20.4450 
20.4695 

74350 
7.4410 
74470 

7-4530 
7-4590 
7.4650 

7.4710 
7.4770 
74829 

129600 

46656000 

18.9737 

7.1138 

i  76400 

74088000 

20.4939 

7.4889 

s 

Na 

jji 

1/N 

fN 

N 

N2 

N8 

V/N 

l^N 

185 


420—540 


K 

N3 

N3 

l/N 

fN 

N 

N« 

N8 

l/N 

fx 

420 

176400 

74088000 

20.4939 

7.4889 

480 

230400 

110592000 

21.9089 

7.8297 

421 
422 
423 
424 
425 
426 

427 
428 

429 

177241 
178084 
178929 
179776 
180625 
181476 
182329 
183184 
184041 

74618461 
75151448 
75686967 
76225024 
76765625 
77308776 
77854483 
78402752 
78953589 

20.5183 
20.5426 
20.5670 
20.5913 
20.6155 
20.6398 
20.6640 
20.6882 
20.7123 

74948 
7.5007 
7.5067 
7.5126 
7.5185 
7.5244 
7.5302 
7.5361 
7-5420 

481 
482 
483 
484 
485 
486 

487 

488 

489 

231361 
232324 
233289 

234256 
235225 
236196 

237169 
238144 
239121 

111284641 
111980168 
112678587 

113379904 
114084125 
114791256 
115501303 
116214272 
116930169 

21.9317 
21-954! 
21-9773 
22.0000 
22.0227 
22.0454 
22.0681 
22.0907 
22.1133 

7.8352 
7.8406 
7.8460 

7.8514 
7.8568 

7.8622 

7.8676 
7.8730 
7.8784 

430 

184900 

79507000 

20.7364 

7.5478 

490 

240100 

i  i  7649000 

22.1359 

7.8837 

43i 
432 
433 
434 
435 
436 

437 
438 
439 

185761 
186624 
187489 
188356 
189225 
190096 
190969 
191844 
192721 

80062991 
80621568 
81182737 
81746504 

82312875 
82881856 

83453453 
84027672 

84604519 

20.7605 
20.7846 
20.8087 
20.8327 
20.8567 
20.8806 
20.9045 
20.9284 
20.9523 

7-5537 
7-5595 
7.5654 
7-5712 
7.5770 
7.5828 

7.5886 

7.5944 
7.6001 

491 
492 
493 
494 
495 
496 

497 
498 
499 

241081 
242064 
243049 
244036 
245025 
246016 
247009 
248004 
249001 

118370771 
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119823157 
120553784 
121287375 
122023936 
122763473 
123505992 
124251499 

22.1585 
22.1811 
22.2036 
22.2261 
22.2486 
22.2711 
22.2935 
22.3159 
22.3383 

7.8891 
7.8944 
7.8998 

7.9051 
7.9105 
7.9158 
7.9211 
7.9264 
7.9317 

440 

193600 

85184000 

20.9762 

7.6059 

500 

250000 

125000000 

22.3607 

7.9370 

441 
442 
443 
444 
445 
446 

447 

448 

449 

194481 

195364 
196249 

197136 
198025 
198916 
199809 
200704 
201601 

85766121 
86350888 
86938307 

87528384 
88121125 
88716536 
89314623 
89915392 
90518849 

2I.OOOO 
21.0238 
2I.O476 
21.0713 
2I.O95O 
2I.II87 
21.1424 
21.  l66o 
21.1896 

7.6H7 
7.6174 
7.6232 
7.6289 
7-6346 
7.6403 

7.6460 
7.6517 
7.6574 

501 
502 
503 
504 
505 
506 

507 
508 

509 

251001 
252004 
253009 
254016 
255025 
256036 
257049 
258064 
259081 

125751501 
126506008 
127263527 
128024064 
128787625 
129554216 
130323843 
131096512 
131872229 

22.3830 
22.4054 
22.4277 
22.4499 
22.4722 
22.4944 
22.5167 
22.5389 
22.5610 

7.9423 
7.9476 
7.9528 
7.9581 
7.9634 
7.9686 

7.9739 
7.9791 

7.9843 

450 

202500 

91125000 

21.2132 

7.6631 

510 

260100 

132651000 

22.5832 

7.9896 

45i 
452 

453 
454 
455 
456 

457 
458 
459 

203401 
204304 
205209 
206116 
2070-25 
207936 
208849 
209764 
210681 

9*733851 
92345408 
92959677 
93576664 
94196375 
94818816 

95443993 
96071912 
96702579 

21.2368 
2I.26O3 
21.2838 
21.3073 
21.3307 
21.3542 

21.3776 
21.4009 
21.4243 

7.6688 
7.6744 
7.6801 

7.6857 
7.6914 
7.6970 
7.7026 
7.7082 
7.7138 

5ii 
512 
513 
5U 
5i5 
516 

5i7 
518 

5i9 

261121 
262144 
263169 
264196 
265225 
266256 
267289 
268324 
269361 

133432831 
134217728 
135005697 

135796744 
136590875 
137388096 
138188413 
138991832 
139798359 

22.6053 
22.6274 
22.6495 
22.6716 
22.6936 
22.7156 
22.7376 
22.7596 
22.7816 

7.9948 

8.0000 
8.0052 
8.0104 
8.0156 
8.0208 
8.0260 
8.0311 
8.0363 

460 

211600 

97336000 

21.4476 

7.7194 

520 

270400 

140608000 

22.8035 

8.0415 

461 
462 
463 
464 

465 
466 

467 
468 
469 

212521 
213444 
214369 
215296 
216225 
217156 
218089 
219024 
219961 

97972181 
98611128 
99252847 
99897344 
100544625 
101194696 
101847563 
102503232 
103161709 

21.4709 
21.4942 
21.5174 

21.5407 
21.5639 
21.5870 

2I.6I02 
21.6333 
21.65  64 

7.7250 
7.7306 
7.7362 
7.7418 
7.7473 
7.7529 

7.7584 
7.7639 

7.769! 

521 
522 
523 
524 
525 
526 

527 
528 

529 

271441 
272484 
273529 
274576 
275625 
276676 
277729 
278784 
279841 

141420761 
142236648 
143055667 
143877824 
144703125 
I4553I576 
146363183 
147197952 
148035889 

22.8254 
22.8473 
22.8692 
22.8910 
22.9129 
22.9347 
22.9565 
22.9783 
23.0000 

8.0466 
8.0517 
8.0569 
8.0620 
8.0671 
8.0723 
8.0774 
8.0825 
8.0876 

470 

220900 

103823000 

21.6795 

7-7750 

530 

280900 

148877000 

23.0217 

8.0927 

47i 
472 
473 
474 
475 
476 

477 
478 
479 

221841 
222784 
223729 
224676 
225625 
226576 
227529 
228484 
229441 

104487111 
105154048 
105823817 
106496424 
107171875 
107850176 
108531333' 
109215352 
109902239 

2I.7O25 
21.7256 
21.7486 
21.7715 
21.7945 
2I.8I74 
21.8403 
21.8632 
2I.886I 

7.78o§ 
7.7860 
7.7915 
7.7970 
7.8025 
7.8079 

7.8I34 
7.8188 
7.8243 

53i 
532 
533 
534 
535 
536 

537 
538 
539 

281961 
283024 
284089 
285156 
286225 
287296 
288369 
289444 
290521 

149721291 
150568768 
151419437 
152273304 
153130375 
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154854153 
155720872 
156590819 

23.0434 
23.0651 
23.0868 
23.1084 
23.1301 
23-1517 

23.1733 
23.1948 
23.2164 

8.0978 
8.1028 
8.1079 
8.1130 
8.1180 
8.1231 
8.1281 
8.1332 
8.1382 

480 

230400 

110592000 

21.9089 

7.8297 

540 

291600 

157464000 

23-2379 

8.1433 

K 

N2 

N3 

1/N 

lTN 

N 

N" 

N3 

1/N 

fN 

136 


540—660 


^ 

N2 

N3 

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N2 

N3 

l/N 

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540 

54i 
542 
543 
544 
545 
546 

547 
548 
549 
550 

55i 

552 
553 
554 
555 
556 

557 
558 
559 
560 

56i 
562 
563 
564 
565 
566 

567 

568 

569 
570 

57i 
572 
573 
574 
575 
576 

577 
578 
579 
580 
581 
582 
583 
584 
585 
586 

587 
588 

589 
590 

59i 

592 
593 
594 
595 
596 

597 
598 
599 
600 

291600 

157464000 

23.2379 

8-1433 

600 
601 
602 
603 
604 
605 
606 

607 
608 
609 
610 
611 
612 
613 
614 

615 
616 

617 
618 
619 
620 
621 
622 
623 
624 
625 
626 

627 
628 
629 
630 

631 
632 
633 
634 
635 
636 

637 
638 

639 
640 
641 
642 
643 
644 
645 
646 

647 

648 

649 
650 

651 

652 

653 
654 
655 
656 

657 
658 

659 
660 

360000 

216000000 

24.4949 

8-4343 

292681 

293764 
294849 

295936 
297025 
298116 
299209 
300304 
301401 

158340421 
159220088 
160103007 
160989184 
161878625 
162771336 
163667323 
164566592 
165469149 

23.2594 
23.2809 
23.3024 

23-3238 
23.3452 
23-3666 

23-3880 
23.4094 
234307 

8.1483 
8-1533 
8.1583 
8.1633 
8.1683 
8.1733 

8.1783 
8.1833 
8.1882 

361201 
362404 

363609 
364816 

366025 

367236 

368449 

369664 
370881 

217081801 
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221445125 
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24.5153 
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8.4390 

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8.4530 
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8.4623 

8.4670 
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302500 

166375000 

23.4521 

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372100 

226981000 

24.6982 

8.4809 

303601 
304704 
305809 
306916 
308025 
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310249 

311364 
312481 

167284151 
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169112377 
170031464 

170953875 
171879616 

172808693 
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174676879 

23-4734 
234947 
23.5160 

23.5372 
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23.5797 
23.6008 
23.6220 
23-6432 

8.1982 
8.2031 
8.2081 
8.2130 
8.2180 
8.2229 
8.2278 
8.2327 
8.2377 

373321 

374544 
375769 

376996 
378225 
379456 
380689 

381924 
383161 

228099131 
229220928 
230346397 
231475544 
232608375 
233744896 
234885113 
236029032 
237176659 

24.7184 
24.7386 
24.7588 
24.7790 
24.7992 
24-8193 

24.839! 
24.8596 
24.8797 

8.4856 
8.4902 
8.4948 
8.4994 
8.5040 
8.5086 

8.5132 
8.5178 
8.5224 

313600 

175616000 

23.6643 

8.2426 

384400 

238328000 

24.8998 

8.5270 

314721 

315844 
316969 

318096 
319225 
320356 

321489 
322624 
323761 

176558481 
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178453547 
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184220009 

23.6854 
23-7065 
23.7276 

23.7487 
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23.8328 
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8.2475 
8.2524 

8.2573 
8.2621 
8.2670 
8.2719 
8.2768 
8.2816 
8.2865 

385641 
386884 
388129 

389376 
390625 
391876 
393129 
394384 
395641 

239483061 
240641848 
241804367 
242970624 
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245314376 
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248858189 

24.9199 
24.9399 
24.9600 
24.9800 
25.0000 
25.0200 
25.0400 

25.0599 
25.0799 

S.53I6 
8.5362 
8.5408 

8-5453 
8.5499 

8.5544 
8.5590 
8.5635 
8.5681 

324900 

185193000 

23.8747 

8.2913 

396900 

250047000 

25.0998 

8.5726 

326041 
327184 
328329 

329476 
330625 
331776 

332929 
334084 
335241 

186169411 
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188132517 
189119224 
190109375 
191102976 
192100033 
193100552 
194104539 

23-8956 
23-9165 
23-9374 

23.9583 
23.9792 
24.0000 
24.0208 
24.0416 
24.0624 

8.2962 
8.3010 
8.3059 
8.3107 
8.3155 
8.3203 

8.3251 
8.3300 
8.3348 

398161 

399424 
400689 

401956 
403225 
404496 

405769 
407044 
408321 

251239591 
252435968 
253636137 
254840104 
256047875 
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259694072 
260917119 

25.1197 
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25.1794 

25.1992 
25.2190 

25.2389 
25.2587 
25.2784 

8.5772 
8.5817 
8.5862 

8.5907 
8.5952 
8-5997 
8.6043 
8.6088 
8.6132 

336400 

195112000 

24.0832 

8.3396 

409600 

262144000 

25.2982 

8.6177 

33756i 
338724 
339889 
341056 
342225 
343396 
344569 
345744 
346921 

196122941 
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200201625 
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24-1039 
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24-2693 

8-3443 
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8-3777 
8.3825 

410881 
412164 
413449 
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417316 
418609 
419904 
421201 

263374721 
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273359449 

25-3180 
25-3377 
25-3574 
25.3772 
25.3969 
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254558 
254755 

8.6222 
8.6267 
8.6312 

8.6357 
8.6401 
8.6446 
8.6490 
8.6535 
8.6579 

348100 

205379000 

24.2899 

8.3872 

422500 

274625000 

254951 

8.6624 

349281 
350464 
351649 
352836 
354025 
355216 

356409 
357604 
358801 

206425071 
207474688 
208527857 
209584584 
210644875 
211708736 
212776173 
213847192 
214921799 

24.3105 
24.33H 
24.3516 

24.3721 
24.3926 
244I3I 
244336 
244540 
244745 

8.3919 
8.3967 
8.4014 
8.4061 
8.4108 
8.4155 
8.4202 
8.4249 
8.4296 

423801 
425104 
426409 

427716 
429025 
430336 
431649 
432964 
434281 

275894451 
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278445077 
279726264 
281011375 
282300416 

283593393 
284890312 
286191179 

25.5147 
25-5343 
25.5539 
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25.6320 

25-6515 
25.6710 

8.6668 
8.6713 
8.6757 
8.6801 
8.6845 
8.6890 
8.6934 
8.6978 
8.7022 

360000 

216000000 

24.4949 

8-4343 

435600 

287496000 

25.6905 

8.7066 

N 

NJ 

N8 

V'N 

fN 

N 

N2 

N8 

1/N 

fN 

137 


660—780 


tf  1  N2 

Ny 

l/N 

f  N 

N 

N2 

N8 

l/N 

?5 

660 
661 
662 
663 
664 
665 
666 

667 
668 
669 
670 
671 
672 
673 
674 

675 
676 

677 

678 

679 
680 
68  1 
682 
683 
684 
685 
686 
687 
688 
689 
690 
691 
692 
693 
694 
695 
696 

697 
698 
699 
700 
701 
702 
703 
704 

705 
706 

707 
708 
709 
710 
711 
712 
7i3 
7i4 
7i5 
716 

717 
718 
719 
720 

435600 

287496000 

25.6905 

8.7066 

720 
721 
722 
723 
724 

725 
726 

727 
728 
729 
730 

73i 
732 
733 
734 
735 
736 

737 
738 
739 
740 
741 
742 
743 
744 
745 
746 

747 

748 

749 
750 

75i 
752 
753 
754 
755 
756 

757 
758 
759 
760 
761 
762 
763 
764 

765 
766 

767 
768 
769 
770 

771 
772 
773 
774 
775 
776 

777 
778 
779 
780 

518400 

373248000 

26.8328 

8.9628 

436921 
438244 
439569 
440896 
442225 
443556 
444889 
446224 
44756i 

288804781 
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296740963 
298077632 
299418309 

25.7099 
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25.8650 

8.7110 

8.7154 
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8.7241 
8.7285 
8.7329 

8-7373 
8.7416 
8.7460 

519841 
521284 
522729 
524176 
525625 
527076 
528529 
529984 
53I44I 

374805361 
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385828352 
387420489 

26.8514 
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26,9815 
27.0000 

8.9670 
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8.9835 
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8.9918 
8.9959 
9.0000 

448900 

300763000 

25.8844 

8.7503 

532900 

389017000 

27.0185 

9.0041 

450241 
451584 
452929 
454276 
455625 
456976 

458329 
459684 
461041 

302111711 
303464448 
304821217 
306182024 
307546875 
308915776 
310288733 
311665752 
313046839 

25.9037 
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25-9615 
25.9808 
26.0000 
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26.0576 

8.7547 
8.7590 
8.7634 

8.7677 
8.7721 
8.7764 
8.7807 
8.7850 
8.7893 

53436i 
535824 
537289 

538756 
540225 
541696 
543169 

544644 
546121 

390617891 
392223168 
393832837 
395446904 
397065375 
398688256 
400315553 
401947272 
403583419 

27.0370 
27.0555 
27.0740 

27.0924 
27.1109 
27.1293 

27.1477 
27.1662 
27.1846 

9.0082 
9.0123 
9.0164 
9.0205 
9.0246 
9.0287 
9.0328 

9.0369 
9.0410 

462400 

314432000 

26.0768 

8-7937 

547600 

405224000 

27.2029 

9.0450 

463761 
465124 
466489 

467856 
469225 
470596 
471969 

473344 
474721 

315821241 
317214568 
318611987 
320013504 
321419125 
322828856 
324242703 
325660672 
327082769 

26.0960 
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26.2107 
26.2298 
26.2488 

8.7980 
8.8023 
8.8066 
8.8109 
8.8152 
8.8194 
8.8237 
8.8280 
8.8323 

549081 
550564 
552049 

553536 
555025 
556516 
558009 
559504 
561001 

406869021 
408518488 
410172407 
411830784 
413493625 
415160936 
416832723 
418508992 
420189749 

27.2213 
27.2397 
27.2580 
27.2764 
27.2947 
27.3130 

27.3313 
27.3496 

27.3679 

9.0491 
9.0532 
9.0572 

9.0613 
9.0654 
9.0694 

9-0735 
9.0775 

9.0816 

476100 

328509000 

26.2679 

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562500 

421875000 

27.3861 

9.0856 

477481 
478864 
480249 
481636 
483025 
484416 
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487204 
488601 

329939371 
331373888 
332812557 
334255384 
335702375 
337153536 
338608873 
340068392 
341532099 

26.2869 
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26.3249 

26.3439 
26.3629 
26.3818 
26.4008 
26.4197 
26.4386 

8.8408 
8.8451 
8.8493 
8.8536 
8.8578 
8.8621 
8.8663 
8.8706 
8.8748 

564001 
565504 
567009 

568516 
570025 
571536 

573049 
574564 
576081 

423564751 
425259008 
426957777 
428661064 
430368875 
432081216 
433798093 
435519512 
437245479 

27.4044 
27.4226 
27.4408 
27.4591 
274773 
27-4955 
27.5136 
2?.53i8 
27-5500 

9.0896 
9.0937 
9.0977 
9.1017 

9-1057 
9.1098 

9-1138 
9.1178 
9.1218 

490000 

343000000 

26.4575 

8.8790 

577600 

438976000 

27.5681 

9-1258 

491401 
492804 
494209 
495616 
497025 
498436 
499849 
501264 
502681 

344472101 
345948408 
347428927 
348913664 
350402625 
351895816 

353393243 
354894912 
356400829 

26.4764 

26.4953 
26.5141 

26.5330 
26.5518 
26.5707 
26.5895 
26.6083 
26.6271 

8.8833 
8.8875 
8.8917 
8.8959 
8.9001 
8.9043 
8.9085 
8.9127 
8.9169 

579121 
580644 
582169 

583696 
585225 
586756 
588289 
589824 
59i36i 

440711081 
442450728 
444194947 

445943744 
447697125 
449455096 
451217663 
452984832 
454756609 

27.5862 
27.6043 
27.6225 
27.6405 
27.6586 
27.6767 
27.6948 
27.7128 
27.7308 

9.1298 
9.*338 
9.1378 
9.1418 
9.1458 
9.1498 

9-1537 
9-1577 
9.1617 

504100 

357911000 

26.6458 

8.9211 

592900 

456533000 

27.7489 

9-I657 

505521 
506944 
508369 

509796 
511225 
512656 
514089 

515524 
516961 

359425431 
360944128 
362467097 
363994344 

365525875 
367061696 

368601813 
370146232 
371694959 

26.6646 
26.6833 
26.7021 
26.7208 

26.739! 
26.7582 

26.7769 
26.7955 
26.8142 

8.9253 
8.9295 

8-9337 
'8.9378 
8.9420 
8.9462 

8.9503 
8.9545 
8.9587 

594441 
595984 
597529 
599076 
600625 
602176 
603729 
605284 
606841 

458314011 
460099648 
461889917 
463684824 
465484375 
467288576 
469097433 
470910952 
472729139 

27.7669 
27.7849 
27.8029 
27.8209 
27.8388 
27.8568 
27.8747 
27.8927 
27.9106 

9.1696 
9.1736 
9-1775 
9.1815 
9-185? 
9.1894 

9.1933 
9.1973 
9.2012 

518400 

373248000 

26.8328 

8.9628 

608400 

474552000 

27.9285 

9.2052 

N  |  N2 

N3 

V'N 

fN 

N    N2 

N3 

l/N 

fN 

13S 


780—900 


5 

N2 

N3 

l/N 

£S 

N 

N2 

N3   |  i/N  |  f  N 

780 

78i 
782 
783 
784 
785 
786 

787 
788 

790 

792 
793 
794 
795 
796 
797 
798 
799 
800 
8oi 
802 
803 
804 
805 
806 

807 
808 
809 
810 
811 
812 
813 
814 

815 
816 

817 
818 
819 
820 
821 
822 
823 
824 
825 
826 

827 
828 
829 
830 

831 
832 

833 
834 
835 
836 

837 
838 

839 
840 

608400 

474552000 

27.9285 

9.2052 

840 
841 
842 
843 
844 

845 
846 

847 
848 
849 
850 

851 
852 
853 
854 
855 
856 

857 
858 
859 
860 
861 
862 
863 
864 
865 
866 
867 
868 
869 
870 
871 
872 
873 
874 
875 
876 

877 
878 

879 
880 
881 
882 
883 
884 
885 
886 
887 
888 
889 
890 
891 
892 
893 
894 
895 
896 

897 
898 

899 
900 

705600 

592704000 

28.9828 

9-4354 

609961 
611524 
613089 
614656 
616225 
617796 

619369 
620944 
622521 

47&37954I 
478211768 
480048687 
481890304 
483736625 
485587656 

487443403 
489303872 
491169069 

27.9464 
27.9643 
27.9821 
28.0000 
28.0179 
28.0357 

28.0535 
28.0713 

28.0891 

9.2091 

9.2130 
9.2170 

9.2209 
9.2248 

9.2287 
9.2326 

9.2365 
9.2404 

707281 
708964 
710649 

712336 
714025 

717409 
719104 
720801 

594823321 
596947688 
599077107 
601211584 
603351125 
605495736 
607645423 
609800192 
611960049 

29.0000 
29.0172 
29-0345 
29.0517 
29.0689 
29.0861 
29.1033 
29.1204 
29.1376 

9-4391 
9.4429 
9.4466 

94503 
9-4541 
9-4578 
9.4615 
94652 
9.4690 

624100 

493039000 

28.1069 

9.2443 

722500 

614125000 

29.1548 

9.4727 

625681 
627264 
628849 
630436 
632025 
633616 
635209 
636804 
638401 

494913671 
496793088 
498677257 
500566184 
502459875 
504358336 
506261573 
508169592 
510082399 

28.1247 
28.1425 
28.1603 
28.1780 
28.1957 
28.2135 
28.2312 
28.2489 
28.2666 

9.2482 

9.2521 
9.2560 

9.2599 
9.2638 
9.2677 
9.2716 

9-2754 
9.2793 

724201 
725904 
727609 

729316 
731025 
732736 

734449 
736164 
737881 

616295051 
618470208 
620650477 
622835864 
625026375 
627222016 
629422793 
631628712 
633839779 

29.1719 
29.1890 
29.2062 
29.2233 
29.2404 
29-2575 
29.2746 
29.2916 
29.3087 

9.4764 
9.4801 
9-4838 

94875 
9.4912 

9-4949 
9.4986 
9-5023 
9.5060 

640000 

512000000 

28.2843 

9.2832 

739600 

636056000 

29.3258 

9.5097 

641601 
643204 
644809 
646416 
648025 
649636 
651249 
652864 
654481 

513922401 
515849608 
517781627 
519718464 
521660125 
523606616 

525557943 
527514112 
529475129 

28.3019 
28.3196 
28.3373 

28.3549 
28.3725 
28.3901 
28.4077 
28.4253 
28.4429 

9.2870 
9.2909 
9.2948 

9.2986 

9-3025 
9.3063 
9.3102 
9.3140 
9.3179 

741321 
743044 
744769 
746496 
748225 
749956 
751689 
753424 
755i6i 

638277381 
640503928 
642735647 
644972544 
647214625 
649461896 
651714363 
653972032 
656234909 

29.3428 

29-3769 

29-3939 
29.4109 
29.4279 
29.4449 
29.4618 
29.4788 

9-5I34 
9-5I7I 
9.5207 

9.5244 
9.5281 
9.5317 
9-5354 
9-5391 
9.5427 

656100 

531441000 

28.4605 

9.3217 

756900 

658503000 

29.4958 

9.5464 

657721 
659344 
660969 

662596 
664225 
665856 
667489 
669124 
670761 

533411731 
535387328 
537367797 
539353144 
541343375 
543338496 

545338513 
547343432 
549353259 

28.4781 
28.4956 
28.5132 
28.5307 
28.5482 
28.5657 
28.5832 
28.6007 
28.6182 

9.3255 
9*3294 
9*3332 

9-3370 
9.3408 

9-3447 

9.3523 
9-356i 

758641 
760384 
762129 

763876 
765625 
767376 
769129 
770884 
772641 

660776311 
663054848 
665338617 
667627624 
669921875 
672221376 

674526133 
676836152 
679151439 

29.5127 
29.5296 
29.5466 

29.563! 
29.5804 

29.5973 
29.6142 
29.6311 
29.6479 

9-5501 
9-5537 
9-5574 
9.5610 

9-5683 

9.5719 
9-5756 
9-5792 

672400 

551368000 

28.6356 

9-3599 

774400 

681472000 

29.6648 

9.5828 

674041 
675684 
677329 

678976 
680625 
682276 
683929 
685584 
687241 

553387661 
555412248 
557441767 
559476224 
561515625 
563559976 
565609283 
567663552 
569722789 

28.6531 
28.6705 
28.6880 
28.7054 
28.7228 
28.7402 
28.7576 
28.7750 
28.7924 

9-3637 
9-3675 
9.3713 
9-3751 

9.3827 

9-3865 
9.3902 
9.3940 

776161 
777924 
779689 
781456 
783225 
784996 
786769 
788544 
790321 

683797841 
686128968 
688465387 
690807104 
693154125 
695506456 

697864103 
700227072 
702595369 

29.6816 
29.6985 
29.7153 
29.7321 
29.7489 
29.7658 
29.7825 

29.7993 
29.8161 

9-5865 
9.5901 
9-5937 

9-5973 
9.6010 
9.6046 
9.6082 
9.6118 
9-6154 

688900 

571787000 

28.8097 

9-3978 

792100 

704969000 

29.8329 

9.6190 

690561 
692224 
693889 

695556 
697225 
698896 
700569 
702244 
703921 

573856191 
575930368 
578009537 
580093704 
582182875 
584277056 
586376253 
588480472 
590589719 

28.8271 
28.8444 
28.8617 
28.8791 
28.8964 
28.9137 
28.9310 
28.9482 
28.9655 

9.4016 

9-4053 
9.4091 

9.4129 
9.4166 
9.4204 
9.4241 
9.4279 
9.4316 

793881 
795664 
797449 
799236 
801025 
802816 
804609 
806404 
808201 

707347971 
709732288 

712121957 
714516984 

716917375 
719323136 

721734273 
724150792 
726572699 

29.8496 
29.8664 
29.8831 
29.8998 
29.9166 
29-9333 
29.9^00 
29.9666 
29.9833 

9.6226 
9.6262 
9.6298 

9-6334 
9.6370 
9.6406 
9.6442 
9.6477 
9.6513 

705600 

592704000 

28.0828 

9-4354 

810000 

729000000 

30.0000 

9-6549 

K 

N' 

N3 

1/N 

£H 

N 

N8   |   N8 

V/N 

?* 

139 


900—1020 


N 

N2 

Ns 

V/N 

fN 

N 

N» 

w 

V/N 

fN 

900 
901 
902 
903 
904 

9°5 
906 

907 
908 
909 
910 
911 
912 
9*3 
914 

9*5 
916 

917 
918 
919 
920 
921 
922 
'923 
924 
925 
926 

927 
928 
929 
930 

93i 
932 
933 
934 
935 
936 

937 
938 
939 
940 

94i 
942 
943 
944 
945 
946 

947 
948 
949 
950 

95i 
952 
953 
954 
955 
956 

957 
958 
959 
960 

810000 

729000000 

30.0000 

9.6549 

960 
961 
962 
963 
964 
965 
966 

967 
968 
969 
970 
971 
972 
973 
974 
975 
976 

977 
978 

979 
980 
981 
982 
983 
984 

985 
986 

987 
988 
989 
990 
991 
992 
993 
994 
995 
996 
997 
998 
999 
1000 

IOO1 
1002 
1003 
IOO4 
1005 
1006 
1007 
1008 
1009 

1010 

IOII 
IOI2 
1013 
IOI4 
1015 

1016 
-1017 
1018 
1019 
1020 

921600 

884736000 

30.9839 

9.8648- 

811801 
813604 
815409 
817216 
819025 
820836 
822649 
824464 
826281 

731432701 
733870808 
736314327 
738763264 
741217625 
743677416 
746142643 
748613312 
751089429 

30.0167 
30.0333 
30.0500 
30.0666 
30.0832 
30.0998 
30.1164 
30.1330 
30.1496 

9-6585 
9.6620 
9.6656 
9.6692 
9.6727 
9-6763 
9.6799 
9.6834 
9.6870 

923521 
925444 
927369 

929296 
931225 
933156 
935089 
937024 
938961 

887503681 
890277128 

893056347 
895841344 
898632125 

gof428696 
904231063 
907039232 
909853209 

31.0000 
31.0161 

31.0322 
31.0483 
31.0644 
31.0805 
31.0966 
31.1127 

31.1288 

9.8683  1 
9.8717 
9.8751 
9.8785 
9.8819 
9-8854. 
9.8888 
9.8922 
9.8956 

828100 

753571000 

30.1662 

9-6905 

940900 

912673000 

31.1448 

9.8990 

829921 
831744 
833569 
835^96 
837225 
839056 
840883 
842724 
844561 

756058031 
758550528 
761048497 
763551944 
766060875 
768575296 
771095213 
773620632 
776151559 

30.1828 
30.1993 
30.2159 
30.2324 
30.2490 
30.2655 
30.2820 
30.2985 
30.3150 

9.6941 
9.6976 
9.7012 
9.7047 
9.7082 
9.7118 

9-7153 
9.7188 
9.7224 

942841 
944784 
946729 
948676 
950625 
952576 
954529 
956484 
958441 

915498611 
918330048 
921167317 
924010424 
926859375 
929714176 

932574833 
935441352 

938313739 

31.1609 
31.1769 
31.1929 

31.2096 
31.2250 
31.2410 

31.2570 
31.2730 
31.2890 

9.9024 
9.9058 
9-9092  / 
9.9126' 
9.9160 
9.9194 
9.9227 
9.9261 
9.9295 

846400 

778688000 

30.3315 

9-7259 

960400 

941192000 

31.3050 

9.9329 

848241 
850084 
851929 

853776 
855625 
857476 

859329 
861184 
863041 

781229961 
783777448 
786330467 
788889024 
7914-53125 
794022776 
796597983 
799178752 
801765089 

30.3480 

30.3645 
30.3809 

30.3974 
30.4138 
30.4302 

30.4467 
30.463! 
30.4795 

9-7294 
9.7329 

9-7364 
9.7400 

9-7435 
9.7470 

9-7505 
9-7540 
9-7575 

962361 

964324 
966289 

968256 
970225 
972196 
974169 
976144 
978121 

944076141 
946966168 
0,  949862087 

952763904' 
955671625 
958585256 
961504803 
964430272 
967361669 

31.3209 
31.3369 
31.3528 

31.3688 
31.3847 

31.4006 

31.4166 

31-4325 
31.4484 

9-9363 
9.9396 
9.9430 

9-9464 
9-9497 
9-9531 
9-9565 
9-9598 
9.9632 

864900 

804357000 

30.4959 

9.7610 

980100 

970299000 

31.4643 

9.9666 

866761 
868624 
870489 
872356 
874225 
876096 

877969 
879844 
881721 

806954491 
809557568 
812166237 
814780504 
817400375 
820025856 
822656953 
825293672 
827936019 

30.5123 
30.5287 
30.5450 

30.5614 
30.5778 
30.5941 
30.6105 
30.6268 
30.6431 

9-7645 
9.7680 

9-7715 
9-7750 
9-7785 
9.7819 

9-7854 
9.7889 

9-7924 

K)8206l 

984064 
986049 
988036 
990025 

992016 

994009 
996004 
998001 

973242271 
'  97619148$ 
979146657 
982107784 
985074875 
988047936 
9910.26973 
994011992 
997002999 

31.4802 

31.4960 
31.5119 
'31.5278 
31.5436 
31.5595 
31.5753 

3I-5911 
31.6070 

9-9699 
9-9733 
9.9766 

,  9.9800 

9-9833 
9.9866 

9.9900 

9-9933 
9.9967 

883600 

830584000 

30.6594 

9-7959 

I  OOOOOO 

IOOOOOOOQO 

31.6228 

10.0000 

885481 
887364 
889249 
891136 
893025 
894916 
896809 
898704 
900601 

833237621 
835896888 
838561807 
841232384 
843908625 
846590536 
849278123 
851971392 
854670349 

30-6757 
30.6920 
30.7083 
30.7246 
30.7409 
30.7571 

30.7734 
30.7896 
30.8058 

9-7993 
9.8028 
9.8063 

9.8097 
9-8132 
9.8167 

9.8201 
9.8236 
9.8270 

I  00200  I 

1004004 
1006009 
1008016 
1010025 
1012036 
1014049 
1016064 
1018081 

1003003001 
1006012008 
1009027027 
1012048064 
1015075125 
1018108216 
1021147343 
1024192512 
1027243729 

31.6386 

31.6544 
31.6702 

31.6860 
31-7017 
3L7I75 

3L7333 
31.7490 
31.7648 

10.0033 
10.0067 

IO.OIOO 

10.0133 
10.0166 

IO.O2OO 

10.0233 
10.0266 
10.0299 

902500 

857375000 

30.8221 

9-8305 

IO2OIOO 

10-30301000 

31-7805 

10.0332 

904401 
906304 
908209 
910116 
912025 
9*3936 
915849 
917764 
919681 

860085351 
862801408 
865523177 
868250664 
870983875 
873722816 
876467493 
879217912 
881974079 

30.8383 
30.8545 
30.8707 
30.8869 
30.9031 
30.9192 

30-9354 
30.9516 
30.9677 

9-s339 
9-8374 
9.8408 

9.8443 
9.8477 
9.8511 
9.8546 
9.8580 
9.8614 

1O22I2I 
I024I44 
1026169 
1028196 
1030225 
'  1032256 
1034289 
1036324 
1038361 

1033364331 
1036433728 
1039509197 
1042590744 
1045678375 

1048772096 
1051871913 

1054977832 

1058089859 

31.7962 
31-8119 

31.8277 

31.8434 
31.8591 
31.8748 
31.8904 
31.9061 
31.9218 

10.0365 
10.0398 
10.0431 
10.0465 
10.0498 
10.0531 
10:0563 
10.0596 
10.0629 

921600 

884736000 

30.9830 

0.8648 

IO4O4OO 

1061208000 

31-9374 

10.0662 

N 

N2 

N3  • 

V/N 

fN 

X 

w 

w 

•  v/N 

fN 

140 


VII 
TABLE  OF  FACTORS 


FOR 


COMPUTING  PROBABLE  ERRORS 


n 

.6745 

j  -6745 

•6745 

l  -6745 

.6745 

}  .6745 

.6745 

!  -6745 

V'n(n—  1) 

l/n(n—  1) 

1/n—  i 

1/n—  l 

V  n(w—  1) 

Vn(n—  1) 

Vn  —  1 

1/n-i 

40 

0.0171 

8.23241 

0.1080 

9-03344 

2 

3 
4 

6 

8 
9 

0.4769 
0.2754 
0.1947 
0.1508 
0.1231 
0.1041 
0.0901 
0.0795 

9.67846 
9.43990  • 
9.28938 
9.17846 
9.09041 

9-01735. 

8.95488 
8.90031 

0.6745 
0.4769 
0.3894 
0.3372 
0:5016 
0.2754 
0.2549 
0.2385 

9.82898 
9.67846 
9.59041  . 
9-52795 
9-47949 
9.43990 
9.40643 
9-37743 

4i 

42 
43 
44 
45 
46 

47 
48 

49 

0.0167 
0.0163 
0.0159 
0.0155 
0.0152 
0.0148 

0.0145 
0.0142 
0.0139 

8.22155 
8.21096 
8.20062 
8.19051 
8.18064 
8.17099 
8.16155 
8.15231 
8.14326 

0.1066 
01053 
0.1041 
0.1029 
0.1017 
0.1005 
0.0994 
0.0984 
0.0974 

9.02795 
9.02258 
9-01735 
9.01224 
9.00725 
9.00237 
8.99760 
8.99283 
8.98835 

10 

0.0711 

8.85185 

0.2248 

9-35I85 

50 

0.0136 

8.13439 

0.0964 

8.98388 

ii 

12 
13 
14 
15 

16 

17 
18 
19 

0.0643 
0.0587 
0.0540 
0.0500 
0.0465 
0.0435 
0.0409 
0.0386 
0.0365 

8.80828 
8.76869 
8.73241 
8.69894 
8.66787 
8.63887 
8.61169 
8.58611 
8.56196 

0.2133 
0.2029 
0.1947 
0.1871 
0.1803 
0.1742 
0.1686 
0.1636 
0.1590 

9.32898 
9.30828 
9-28938 
9.27200 

9-2559I 
9.24093 

9.22692 

9-21375 
9.20134 

5i 

52 
53 
54 
55 
56 

57 
58 
59 

0.0134 
0.0131 
0.0128 
0.0126 
0.0124 

O.OI22 
O.OII9 
O.OII7 
O.OII5 

8.12571 
8.11719 
8.10884 
8.10064 
8.09260 
8.08470 
8.07694 
8.06932 
8.06184 

0.0954 
0.0944 
0.0935 
0.0926 
0.0918 
0.0900 
•0.0901 
0.0893 
0.0886 

8.97949 
8.97519 
8.97097 
8.96684 
8.96278 
8.95879 
8.95488 
8.95104 
8.94726 

20 

0.0346 

8.53908 

0.1547 

9,18960 

60 

O.OII3 

8.05447 

0.0878 

8-94355 

21 

22 
23 
24 

25 
26 

27 
28 
29 

0.0329 
0.0314 
0.0300 
0.0287 
0.0275 
0.0265 
0^255 
00245 
0.02.37 

8-5*735 
8.49665 
8.47690 

8.45801 
8.43990 
842252 
8.40581 
8.38971 
8.37420 

0.1508 
0.1472 
0.1438 
0.1406 

0.1377 
0.1349 

0.1323 
0.1298 
0.1275 

9.17846 
9.16787 
9.15776 
9.14811 
9.13887 
9.13001 
9.12149 
9.11329 
9.10540 

61 
62 
63 
64 
65 
66 

67 

68 
69 

O.OIII 
O.OIIO 
O:OIO8 

0.0106 
0.0105 
O&IO3 

O.OIOI 
O.OIOO 

0.0098 

8.04723 
8.04011 
8.03311 
8.02622 
8.01943 
8.01275 
8.00617 
7.99968 
7-99330 

0.0871 
0.0864 
0.0857 
0.0850 
0.0843 
0.0837 
0.0830 
0.0824 
0.0818 

8.93990 
8.93631 
8.93278 
8.92931 
8.92589 
8.92252 
8.91920 
8.91594 
8.91272 

30 

0.0229 

8.35922 

0.1252 

9.09778 

70 

0.0097 

7.98700 

0.0812 

8.90955 

3i 
32 
33 
34 
35 
36 

37 
38 
39 

O.O22I 
0.02  1  4 
Q.0208 
O.O2OI 
0.0196 
O.OI9O 
O.OI85 
0.0180 
0.0175 

8-34473 
8.33072 
8.31714 
8.30398 
8.29120 
8.27879 
8.26672 
8.25498 
8.24355 

0.1231 

O.I2II  . 
O.II92 

o.i  1  74 
0.1157 
0.1140 
0.1124 
0.1109 
0.1094 

9.09041 
9.08329 
9.07640 
9.06972 
9-06324 
9.05694 
9.05082 
9.04487 
9.03908 

7i 

72 

73 

74 
75 
76 

77 
78 

79 

0.0096 

0.0094 
0.0093 

0.0092 
0.0091 
0.0089 
0.0088 
0.0087 
0.0086 

7.98080 
7.97468 
7.96865 
7.96270 
7-95683 
7.95104 

7-94532 
7.93968 
7-934U 

0.0806 
0.0800 
0.0795 
0.0789 
0.0784 
0.0779 
0.0774 
0.0769 
0.0764 

8.90643 

8.90335 
8.90031 

8.89731 
8.89436 
8.89144 
8.88857 
8.88573. 
8.88293 

40 

O.OI7I 

8.23241 

0.1080 

9-03344 

80 

0.0085 

7.92962 

0.0759 

8.88016 

.6745  . 

j  .6745 

.6745 

j  .6745 

.6745 

j  .6745 

.6745 

!  -6745 

V7n(n-l) 

l/n(n—  1) 

V/n-1 

1/n  —  1 

/n(.n—  1) 

V/nn—  1) 

/«-! 

y  n—  i 

FORMULAS. 


GENERAL  TRIGONOMETRIC  FORMULAS. 


sin  2a 

sin  («  ±  /3)  =  sin  a  cos  /?  ±  cos  a  sin  /? . 

COS  (a  ;b  /3)  =  COS  a  COS  /3  =p  Sin.  a  gin  /?. 

tan  a  ±  tan  /? 


sin  2  a        =2  sin  a  cos  a . 

COS  2  a         =COS  2a— sin  2a  =  1  —  2sln'a=  2  COS  "a-— 1. 

2  tana 
**u2a       ^l^tiiPa- 

sin2 «         =H1— cos2a). 

COS  2  a  =  J  (1  -|-  COS  2  a). 

sin  2a 


sin  a  +  sin  0  =  2  sin  J  (<H-  0)  cos  f  (a  —  /?). 
sin  a  —sin  y?  =  2  cos  $  (a  4-  (3)  sin  £  (a  —  ^3). 
cos  a  -f  cos  £  =  2cosJ(a-f-/9)  cos  J(o—  ^). 
cos  j3—  cos  o  =  2sm  J 


sin  20—  sin  2/3  =  cos  2/3—  cos  *«  =  sin  (a  -f  £}  sin  (a—  /S). 
cos  2a—  sin  2^  =.  cos  (a  -j-  j8)  cos  (a—p). 


FORMULAS  FOR  PLANE  TRIANGLES* 

In  these  formulas  «,  b  and  c  denote  the  sides  and  A,  B  and  C  the  opposite 
angles.  1C  denotes  the  area  and  s  =s  %  (a  -f-  6  +  c).  Only  one  formula  of  each  set  is 
given,  the  other  two  may  be  obtained  by  advancing  the  letters. 

(21)  a     -  --—  =•     ° 
sin  A      sin  B     sin  G  ' 

(22)  o  +  6  =  ton  H*  +  3) 
a—  6     ton  J  (4  —3). 

(23)  a2       ==5»_j_c2_ 

(24)  o        =  6  cos  C  +  c  cos 


60 

exit 


142  FORMULAS. 

(26)  cosM  =  ^|^. 


(50) 


(s—  a) 


(27)  tenM 

(28)  JT=  £  <z  6  sin  (7  =  is  (s—  a)  (5  —  6)(s—  c). 

FORMULAS  FOB  RIGHT  SPHERICAL   TRIANGLES. 

Benoting  the  'right  angle  by  C,  the  formulas  are 

(29)  sin  a  =  sin  J.  sin  c. 

(30)  sin  6  =  sin  B  sin  c. 

(31)  tan  a  —  cos  J?  tan  c  =  tan  J.  sin  6. 

(32)  tan  b  =  cos  A  tan  c  =  tan  B  sin  a. 

(33)  cos  A  =  cos  a  sin  B. 

(34)  cos  5  =  cos  6  sin  A. 

(35)  cos  cr  =  cos  a  cos  6. 

(36)  cos  c  =-  cot  J.  cot  B. 

FORMULAS  FOR  THE  GENERAL   SPHERICAL   TRIANGLE. 

cos  a  =  cos  6  cos  e  +  sin  6  sin  c  cos  A. 

(37)  sin  a  sin  B  =  sin  6  sin  J.. 

(38)  sin  a  cos  J5  =  cos  b  sin  c  —  sin  b  cos  c  cos  A. 

(39)  sin  a  cos  C  =  cos  c  sin  b  —  sin  c  cos  b  cos  J.. 

(40)  si-n  A  cot  J?  =  cot  6  sin  c  —  cos  c  cos  A 

(41)  Sin  A  cot  C  ==  cot  c  sin  6  —  cos  6  cos  A. 

(42)  sin  A  cos  -6  —  cos  B  sin  (7  +  sin  B  cos  (7  cos  a. 

(43)  sin  A  cos  c  —  cos  C  sin  _B  -f  sin  C  cos  .#  cos  a. 

(44)  *sin  a  cot  b  =  cot  _Z?  sin  C  -f  cos  (7  cos  a. 

(45)  sin  a  cot  c  =  cot  (7  sin  B  +  cos  5  cos  a. 

(46)  cos  J.  =  sin  B  sin  (7  cos  a  —  cos  B  cos  C. 

Putting  a  =  J(a  +  &+c)  and  fl- 


sin  B  sin  C 

Uffnn 

(51)  cos  £  a  = 


sin  B  sin 
(52)  tan  \  a  = 


(53)  sin  M  sin  ^  (6  +  .c)  =  ± sin  i  a  cos  J  (5  —  C). 

(54)  sin  £  J.  cos  J  (6  +  c)  =  =h  cos  J  a  cos  J  (5  +  C7). 

(55)  cos  J  A  sin$(6— c)  —  dfc  sin  }asin  *  (B  —  C). 

(56)  cos  M  cos  J  (6  — c)  =  rb  cos  J  a  sin  }  (B  +  C). 

(57)  tan*  J  1T=  tan  Js  tan  *  (5— a)  tan  }  (5—6)  tan  J  (s— c). 


FORMULAS.  143 

FORMULAS  RESULTING  FROM  THE  METHOD  OF  LEAST  SQUARES. 
Formulas  for  Combining  Observations  and  Determining  Probable  Errors. 

1.  Direct  observations  of  a  quantity:   n  separate  results,  ml,mt}  ...mn  of 
equal  weight. 

Most  probable  value  of  quantity,  z  =~  —  . 
Residuals, z  —  ml  =  ^ ,  z—  m^  =  v2>... z  —  m*  —  vn . 

Probable  error  of  z ,  rQ  =  ± 0.6745 A /  ^vv^    . 

\  n  (n—l) 

Probable  error  of  a  single  observation,  r  =  ±  0.6745  A I  [^3.  . 

\  n  —  l 

2.  Direct  observations  of  a  quantity:   n  separate  results,  ml ,  w2 ,  . . .  mn  of 
unequal  weights,  Pi,p2,  . . . pn • 

Most  probable  value  of  quantity,  z  =  \-Pm *.. 


Probable  error  of  z ,  rQ  —  db  0.6745 

Probable  error  of  an  obs'n  of  weight  unity,   r  =  ± 0.6745       [Pvv~\  t 

\    n  —  1 

Weight  of  z,  P  =  [p~\> 

Relation  of  weights  to  probable  errors,          pl  :p2 :  . .  : :  — j  :  — ^  :  . . 

3.  If  Z  =  azl  ±bz.2±:  ...  JtZn ,  and  the  probable  errors  and  weights  of  z,  ^z^...zn 
are  rlf  r2,  . . .  rn  and  pltp3i  ...pn,  then  the  probable  error  and  weight  of  Z  are  given 
by 


P  Pl  P2  '    Pn' 

4.     In  general,  if  Z  =f  (zl ,  z2 ,  . . .  zn),  the  probable  error  of  Z  is 

r  =  ±. 


5.  Direct  observations  of  a  function  of  a  quantity  z:  the  separate  results, 
wx ,  m, , . . .  mn  of  equal  weight,  and  the  form  of  the  function,  az.  The  observation 
equations  are 


The  most  probable  value  of  z  and  its  probable  error  are 

^_[om]  r  =  ±0.6745  J i>^ 

[oo]  \  [oo]  (?i -1). 

If  the  observations  are  of  unequal  weights,  multiply  the  observation  equations 
through  by  the  square  roots  of  their  respective  weights,  and  proceed  as  before. 

6.    Direct  observations  of  a  function  of  twro  quantities,  w  and  z  :  the  separate 

*The  symbols  [  ]  signify  the  sum  of  all  similar  quantities.    Thus, 
[  m  ]  EE  wii  +  W2  +  . . .  +  m«  . 


144  FORMULAS. 

results,  Wu  ra2,  . . .  mn  of  equal  weights,  and  the  form  of  the  function,  aw-{-  bz.    The 
observation  equations  are 

ax  w  +  &i  2  +  ??h  =  0 , 

—  0 , 


an  w  +  bn  z  +  w«  =  0  . 
The  normal  equations  are 


Let 


Then  the  most  probable  values  of  w  and  z  are  given  by 

[am] 


^  , 

[aa]        [  aa  ] 


The  weights  of  w  and  z  are 


The  probable  error  of  a  single  observation  (of  weight  unity)  is 

r  =  io 

and  the  probable  errors  of  ^  and  ^  are 


If  the  observations  are  of  unequal  weights,  multiply  the  observation  equations 
through  by  the  square  roots  of  their  respective  weights  and  proceed  as  before. 

7.  Direct  observations  of  a  function  of  three  quantities,  x,  y  and  z\  the 
separate  results.  m1,m2....mw  of  equal  weight,  and  the  form  of  the  function, 
ax  -\-~by-\-  cz.  The  observation  equations  are 

,  =0, 
0  -f  ra2  =  0  , 


»  y  +  cnz  +  mtt  =  0  . 
The  normal  equations  are 


Let 


=£cca], 


FORMULAS.  145 


Then  the  most  probable  values  of  x,  y  and  z  are  given  by 

[CM.2] 


-          [JTO.1] 

[66.1]  [6  6.1]' 

Q6  ]  ^      [acl^ 

37  —  r  ,     H  -~  f r  •%•  •  i 

[a  a  ]          [aa]        [a  a  ] 


The  weights  of  x,  y  and  z  are  given  by 

jp.  =[cc.2], 


in  which 


The  probable  error  of  a  single  observation  (of  weight  unity)  is 

r  =±  0.6745 


—  3 
and  the  probable  errors  of  a?,  #  and  z  are 


-  r  -  r  - 

—      /  -  ,       ly  —       /  -  f       lz  —       /  — 
m  V  Py  V  PZ 


If  the  observations  are  of  unequal  weights  multiply  the  observation  equations 
through  by  the  square  roots  of  their  respective  weights,  and  proceed  as  before. 


CONSTANTS. 


Mathematical  and  Astronomical  Constants. 

Base  of  natural  logarithms e  =  2.71828183 

Modulus  of  common  logarithms //  =  0.43429448 

Radius  of  a  circle  in  degrees f  =  57.29578 

"         "       "      "  minutes ?*  =  3437.7468 

"         "       "      "  seconds r  =  206264.806 

Circumference  of  a  circle  in  degrees c  =  360 

"  "        "      "  minutes  ...  0  =  21600 


Sine  of  one  second 


seconds c  =  1296000 

• 0.000004848137 

^  =  3.14159265 

-  =0.31830989 
^  =  9.86960440 

1/7T  =  1.77245385 


log. 

0.43429448 

9-63778431 
1.75812263 
3-53627388 
5.31442513 
2.55630250 

4.334453/5 
6.11260500 
4.68557487 
0.49714987 

9.50285013 
0.99429975 
0.24857494 


*/^ 

-y  £-=  0.80599598  9.90633287 

Mean  solar  days  in  a  Julian  year 365.25  2.5625902 

"        "        "       "    sidereal  "..... 365.25637  2.5625978 

"        "        "       "    tropical  "      365.24222  2.5625809 

"         "        "       "    sidereal  day 0.99726957  9.9988126 

Sidereal         "       "    mean  solar  day 1.00273791         .  0.0011874 

Number  of  seconds  in  a  day 86400  4.9365137 

"        "        "  "    sidereal  year 31558150  7.4991115 

Square  root  of  the  attractive  force  of  the  sun  (Gauss)  k  =  0.01720210  8.235581. 

««        "       "  "  "         "         "    in  see's  k  —  3548.18761  ^. 5500066 

Time  required  for  light  to  traverse  the  distance  from 

the  earth  to  the  sun,  according  to  Struve   .    .    497". 78  2.6970374 

Equatorial  horizontal  parallax,  according  to  Xewcomb  .  8 ".848  0.9468451 

Aberration  constant,  according  to  Struve 2o"445i  1.3105892 

Nutation  constant,  according  to  Peters 9". 2236  +  ©".000009  (t — 1850). 

General  precession,  according  to  Struve 5o".2524-(-o".ooo2268  (t — 1850). 

Precession  constants  for  the  equator,  accord- )      j  in  =  46".o76s  -f- ©".0002849  (£—1850). 
ing  to  Struve  and  Peters,  (tropical  year,) )      |  n  —  20^.0564  —  ©".0000863  (t — 1850). 
Obliquity  of  the  ecliptic,  according  to  Struve    .... 

23°  27'  30". 76  —  0*4738  (t — 1850)  —  ©".0000014  (i — 1850)'. 

Comparison  of  Linear  Measures 

log. 

i  English  inch 0.02539977  metres  8.4048298 

i       "       foot 0.30479727       "  9.4840111 

i        "       yard 0.91439180       "  9.9611323 

i  metre 3.28086933  English  feet  0.5159889 

i  centimetre 0.39370432       "       inches  9.5951702 

i  toise  =  6  Paris  feet 1.94903631  metres  0.2898199 

i  Paris  foot  =  12  Paris  inches 0.32483938       "  9.5116687 

i  Paris  inch  =  12  Paris  lines 0.02706995       "  8.4324874 

i  Paris  line 0.00225583       '  7-3533062 

cxlvi 


CONSTANTS. 


147 


Compression,  p  =       -  =  =  0.003342773 


Dimensions  of  the  Earth  according  to  Bessel. 

Semi-axis  major    .........    a  =  3962.8025  English  miles 

20923597         "       feet 
6377397.15  metres 

Semi-axis  minor    .........    &  =  3949-5557  English  miles 

20853654         "       feet 
6356078.96  metres 

2     *  g8 

Eccentricity  ...........    6  =  0.08169683 

Quadrant  of  a  meridian  ......     Q  =  10000855.76  metres 

Dimensions  of  the  Earth  according  to  Clarke  (1880). 

Semi-axis  major    .........    a  =  6378249.2  metres 

Semi-axis  minor      .........  6^6356515.0       " 

Compression      ....  p  =  293*465  =  aoo34°756 

Eccentricity    ...........  6  =  0.0824831 

Quadrant  of  a  meridian    ......  Q  =  10001869  metres 


log. 

3.5980x324 
7.3206363 
6.8046435 
3.5965482 
7.3191822 
6.8031893 

7.5241069 

8.9122052 
7.0000372 

log. 

6.8047015 
6.8032191 

7-5324435 

8.9163649 
7.0000812 


Constants  for  Reducing  to  and  from  the  C.   G.  S.  System  of  Measures. 


i  inch 
i  foot 
i  yard 
i  mile 


=   2.5400  centimetres. 

=  30-4797 
=  91.4392 
=  160933. 


LENGTH. 

i  cm.  =  0.39370  inches. 
i   «    =  0.032809  feet. 
i   «    =  0.010936  yards. 
i   "    =  6.2138  x  io-6  miles. 


i  naut.  mile  =  185230. 

i  square  inch  =       6.4516  square  cm. 

i  square  foot  =    929.01 

i  square  yard  =  8361.13 

i  square  mile  =  2.59  X  io10      " 

r  cubic  inch  =  16.387  cubic  cm. 
i  cubic  foot  =  28316. 
i  cubic  yard  =  764535.         " 
i  gallon        —  4541.  " 


i  grain        =  0.064799  grams, 
i  oz.  avoir.  =  28.3495      " 


=  5.398  x 


nautical  miles. 


ilb. 


=  453-59 


AREA. 

i  sq.  cm.  =0.1550  square  inches, 
i      "        =  0.001076  square  feet, 
i      "        =0.0001196  square  yards, 
i      "        =  3.861  x  io-11  square  miles. 

VOLUME. 

i  cubic  cm.  =  0.06102  cubic  inches, 
i          «        =  3.532  x  io-5  cubic  feet, 
i          "       =  1.308  x  io -6  cubic  yards, 
i         "       =  0.0002202  gallons. 

MASS. 

i  gram  =  15.432  grains, 
i      "     =  0.035274  oz.  avoir, 
i      "     =  0.0022046  Ib.     " 


VELOCITY  AND  ACCELERATION. 

i  foot  per  sec.         =  30.4797  cm.  per  sec.    i  cm.  per  sec.  =  0.032809  feet  per  sec. 
i  stat  mile  per  hr.  =  44.704         "     "          i        "     "        =  0.022369  stat.  miles  per  hr. 
i  naut.  mile       "       =51.453         "     "          i        "     "        =0.0 1943 5  naut.  miles  per  hr. 
i  km.  per  hour         =  27.7778       "     "          i        "     "       =  0.036  km.  per  hour. 

i  foot  per  sec.  per  sec.  =  30.4797  cm.  per  sec.  per  sec. 

i  cm.  per  sec.  per  sec.  =  0.032809  feet  per  sec.  per  sec. 

DENSITY. 

i  Ib.  per  cubic  foot  =  0.016019  gm.  per  c.  c.    i  gm.  per  c.  c.  =  62.426  Ib.  per  cubic  foot, 
i  gr.  per  cubic  inch  =  0.003954    "     "     "       i    "     "      "    =  252.88  gr.  «       «      inch. 


148.  r  >  CONSTANTS. 

FORCE  IN  ABSOLUTE  MEASURE. 

Weight  of  i  gram          =  981          dynes,      i  dyne  =  weight  of  0.001019  grams. 
"          i  grain         =63.57  "  -.        i     "     =       "          0.01573  grains. 

i  oz.  avoir.  =  2.78  X  io4   "          i     "     =       "          3.597  x  io-5  oz.  avoir, 
ilb.       "      =  445Xio5   "  i     "     =       «  2.247  xio-6lb.     " 

i  poundal  =  13825.  i     "     =  7.2333  X  io~5  poundals. 

(The  ratio  of  the  poundal  to  the  dyne  is  independent  of  g). 

WORK  AND  EN3RGY  IN  ABSOLUTE  MEASURE. 


i  gm.  cm.  =  981          ergs, 

i  kilogrammetre  =  9.81  X  io7  " 
i  foot-pound  =  i.356Xio7  " 
i  foot-poundal  =  421390.  " 
i  joule  =  io7  ergs. 


erg  =  0.001019  gramcentimetres. 
"     =  1.019  X  io-8  kilogrammetres. 
"     —  7-37  X  io-8  foot-pounds. 
"     =  2.3731  X  io-6  foot-poundals. 
«     =  io-7  joules. 


( The  ratio  of  the  foot-poundal  to  the  erg  is  independent  of  g ). 

WORK  IN  GRAVITATION  MEASURE. 

i  foot-ton       —  3.097  x  io7  gm.  cm.       i  gm.  cm.  =  6.494  x  io-8  foot-tons, 
i  foot-pound  =  13825  "  i       "        -  7.2331  X  icr-5  foot-pounds, 

i  foot-grain    =  1.975  i       "       =  0.50632  foot-grains. 

RATE  OF  WORKING  IN  ABSOLUTE  MEASURE. 

i  horse-power  =  7.46  x  io9  ergs  per  sec.  i  erg  per  sec.  =  1.34  X  io-10  horse-power, 
i  f orce-de-cheval  =  7.36  x  io9  "  "  i  "  "  =  1.36  x  io-10  force-de-chev. 
i  watt  =io7  "  "  i  "  "  =  io-7  watts. 

RATE  OF  WORKING  IN  GRAVITATION  MEASURE. 

i  horse-power    =  7.604 Xio6gm.cm.per sec.    i  gm.cm.persec.=i. 3151X10— 7horse-pow. 
i  force-de-chev.=7.5Xio6          "          "         i       "         "        =1.3333X10— 7  f.-cle-chev. 

Other  Physical  Constants. 

cubic  inch  of  pure  water,  at  4°  C,  weighs  252.89  grains, 
cubic  foot  of  pure  water,  at  4°  C,  weighs  62.43  pounds, 
cubic  inch  of  mercury,  at  o°  C,  weighs  3439  grains  =  0.4913  pounds, 
litre  of  dry  air,  at  o°  C,  pressure  760  mm.,  weighs  1.2932  grams, 
cubic  foot  of  dry  air,  at  o°  C,  pressure  760  mm.,  weighs  565.1  grains, 
horse  power  =  550  foot  Ibs.  per  sec.  =  33000  foot  Ibs.  per  miuute. 
Force  of  gravity  at  the  sea  level  for  the  latitude  </>, 
in  metres,  g  =  9.7810  +  0.0503  sin  V; 
in  feet,      g  =  32.0902  +-0.1650  sin  20; 

Length  of  seconds'  pendulum  at  the  sea  level  for  the  latitude  <j>, 
in  metres,  I  =  0.99102+0.00510  sin  V; 
in  inches,  I  =  39.0169  +  0.20080  sin  z<f>. 

Velocity  of  light  in  vacuum,  according  to  Michelson,  ,.  ^    u^  \j^  ^  <^ 

2Q6..944-kmT  per  sec.  =  186378  miles  per  sec. 

Velocity  of  sound  in  air,  

in  metres  per  sec.,  v  =  331.7  i/  1  +  0.003665  t,  where  t  =  degrees  Cent, 
in  feet  "        v=  1088.3 j/  1+0.002036(^-32), "   t  =       "       Fahr. 

Difference  of  elevation, 

in  feet,  II  =  60360  (log  P  —  log  p)  (i  +  -  *"  g~  64) ,  where  P  and  p  are  the 

barometric  heights  in  inches,  and  T  and  t,  the  temperatures  in  degrees 
Fahr.  at  the  lower  and  upper  stations  respectively. 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

ASTRONOMY,  MATHEMATICS- 


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